Algebra Problems of the Day (Algebra Regents, June 2026 Parts III and IV)

This exam was adminstered in June 2026.

June 2026 Algebra, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

31. A ball is tossed up into the air from the deck of a building. The distance that the ball is above the ground t seconds after it is tossed can be modeled by the function D(t) = −16t² + 32t + 48, where the distance is measured in feet. On the set of axes below, graph D(t) = −16t² + 32t + 48.

State the maximum number of feet above the ground that the ball will reach.

State the number of seconds after the toss it will take the ball to hit the ground.

Answer:

Use your graphing calculator to find the points that you need to plot before t = 0 and when h(t) = 0.

You'll get the following points: (0,48), (1,64), (2,48), (3,0). Plot these and draw the curve. Do not draw any arrows at the endpoints.

The ball will reach a maximum height of 64 feet.

The ball will hit the ground at 3 seconds.

32. Solve the following system of equations algebraically for all values of x and y:

y = −2x + 3
y = x² − 5x + 3

Answer:

Set the expressions equal to each other and solve the quadratic equation.

x² − 5x + 3 = -2x + 3
x² − 3x = 0
x(x - 3) = 0
x = 0 or x - 3 = 0
x = 0 or x = 3

Solve for y:

If x = 0, then y = -2(0) + 3 = 3.

If x = 3, then y = -2(3) + 3 = -3

33. The table below shows the number of years of experience, x, working as a salesperson and their corresponding salaries, y, in thousands of dollars.

State the linear regression equation for these data. Round all values to the nearest hundredth.

State the correlation coefficient for this data set, to the nearest hundredth.

State what the correlation coefficient indicates about the linear fit of the data.

Answer:

Put the data into Lists L₁ and L₂. Make sure that "DiagnosticOn" has been executed. (It's in the Catalogue.) Run a Linear Regression.

COPY THE ENTIRE WINDOW OF DATA ONTO YOUR TEST PAPER. The reason for this is if you entered any piece of information incorrectly, all of the following answers are likely to be incorrect. However, if you show the data that you were given by your calculator, you can still earn three of the four points.

The equation for the linear equation will be y = 4.61x + 11.93

The correlation coefficient is r = 0.98.

The correlation coefficient indicates a strong correlation.

There isn't much to say about this. Either you know how to enter the data and run the linear regression, or you do not.

34. Graph the system of inequalities on the set of axes below.

2y < x − 8
3x + y ≥ 6

State the coordinates of a point that satisfies both inequalities. Justify your answer

Answer:

Rewrite the inequalities, solving for y in terms of x and graph.

Remember that < means a borken line that gets shading underneath, and ≥ means a solid line that gets shaded above.

2y < x − 8
y < 1/2 x − 4

3x + y ≥ 6
y ≥ -3x + 6

Your graph should look like this:

Any point in the area labeled "S" is a valid point that satisfies both inequalities. Do NOT choose any point on the solid line.

For example: (8, -2) is in the double-shaded area, marked "S".

June 2025 Algebra, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. At a department store in a tax-free state, Jane can either buy three tank tops and two sweatshirts for $52 or two tank tops and one sweatshirt for $30. If x represents the price of one tank top and y represents the cost of one sweatshirt, write a system of equations that could be used to model this situation. On the set of axes below, graph the system of equations.

State the coordinates of the point of intersection of your lines. Explain what each coordinate of the point of intersection means in the context of the problem.

Answer:

Write the equations from the first statement.

3x + 2y = 52 and 2x + y = 30

Rewrite these in slope-intercept form so you can graph them.

3x + 2y = 52
2y = -3x + 52
y = -3/2 x + 26

2x + y = 30
y = -2x + 30

Put these equations in your graphing calculator and plot the points from the tables of values.

Your graph will look like this.

The point of intersection is (8,14). In the context of the problem, tank tops cost $8 and sweatshirts cost $14.

End of Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
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If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra Problems of the Day (Algebra Regents, June 2026 Part II)

This exam was adminstered in June 2026.

June 2026 Algebra, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. Express the product of (1 − 2x) and (3 − 5x) as a polynomial in standard form.

Answer:

Use the Distributive Property, box method or even "FOIL". Just be sure to put it into standard form when combining like terms.

(1 − 2x) (3 − 5x)
= (1)(3 - 5x) - 2x(3 - 5x)
= 3 - 5x - 6x + 10x²
= 10x² - 11x + 3

26. On the set of axes below, graph f(x) = |x| − 3 over the domain −7 ≤ x ≤ 7.

Answer:

Use your graphing calculator. The abs() function is in the Catalogue.

An absolute value graph will be V shape. The vertex of this graph will be (0,-3). The slope on the left side is -1 and the slope on the right side is 1. The graph will be symmetrical.

Only graph between -7 and 7. The graph should have two endpoints. DO NOT put arrows on the graph. Do not go beyond -7 or 7. Follow instructions.

Your graph will look like this.

27. Solve the inequality algebraically: −4x + 1 > 9 + 3(2x + 1) + x

Answer:

Simplify each expression. Then isolate the variable with inverse operations.

−4x + 1 > 9 + 3(2x + 1) + x
−4x + 1 > 9 + 6x + 3 + x
−4x + 1 > 7x + 12
−11x > + 11
x < 1

28. Solve the formula A = 1/2 bh for h in terms of A and b.

Answer:

You can use a graphing calculator to give you a Five Number Summary of this data by putting it into a List.

A = 1/2 bh
2A = bh
2A / b = h

29. The table below shows the population of Manhattan for the years indicated, according to the U.S. Census Bureau.

Determine the average rate of change of the population per year between the years 1980 and 2020, rounded to the nearest integer.

Answer:
The average rate of change between the years 1980 and 2020 is just the slope of the line connecting those two ordered pairs.

(1,643,734 - 1,428,285) / (2020 - 1980) = 5386.225, which rounds to 5386.

30. Rewrite 5 / √(3) as a fraction with a rational denominator.

Answer:
Rationalize the denominator by multiplying by √(3) / √(3), which is equal to 1. (Multiplying a fraction by 1 doesn't change its value.)

5 / √(3) X √(3) / √(3) = 5√(3) / 3

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra Problems of the Day (Algebra Regents, June 2026 Part I)

This exam was adminstered in June 2026.

More Regents problems.

June 2026 Algebra Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


1. What is the 20th term of the arithmetic sequence 4, 7, 10, 13, ...?

(1) 61
(2) 64
(3) 79
(4) 83

Answer: (1) 61

The Common Difference is 7 - 4 = 3. The 20th term is 19 X 3 more than the first term.

4 + 19 X 3 = 61, which is Choice (1).

2. What is the value of x in the equation 0.5x − 4 = 8 − x?

(1) 6
(2) 8
(3) 18
(4) 24

Answer: (2) 8

Isolate the variable by adding x to both sides of the equation and then use inverse operations.

0.5x − 4 = 8 − x
1.5x − 4 = 8
1.5x = 12
x = 8

Choice (2) is the correct answer.

3.The binomial 4x² − 25 is equivalent to

(1) 4(x+5)(x−5)
(2) 4(x−5)(x−5)
(3) (2x−5)(2x+5)
(4) (2x−5)(2x−5)

Answer: (3) (2x−5)(2x+5)

The Difference of Squares Rule applies. Every part of this binomial is a perfect square, so the binomial can be broken down into two conjugates.

4x² − 25 = (2x - 5)(2x + 5), which is Choice (3).

4. A function is defined by the following set of points: {(3,−4),(−4,3),(1,1),(x,2)}
What is a possible value for x?

(1) 1
(2) 2
(3) 3
(4) -4

Answer: (2) 2

For a relation to be a function, no input value can repeat with a different output value. The numbers 1, 3, and -4 are already in this function and can be eliminated.

That leave only Choice (2) 2, which is the correct answer.

The fact that 2 appears in the output doesn't matter.

5. The expression (4xy²)³ is equivalent to

(1) 12x³y⁶
(2) 12x³y⁸
(3) 64x³y⁶
(4) 64x³y⁸

Answer: (3) 64x³y⁶

The exponent of 3 is applied to every coefficient and variable inside the parentheses. The exponents are multiplied. If they don't have an exponent, then the exponent is 1.

(4xy²)³ = 4³x³y^(2*3) = 64x³y⁶

This is Choice (3).

6. Allison was asked to write a third-degree trinomial with a leading coefficient of 4 and a constant term of 5. Which expression satisfies these conditions?

(1) 4x³ - 5
(2) 3x⁴ + 5
(3) 4x³ + 8x² + 5
(4) 3x⁴ + 2x³ - 5

Answer: (3) 4x³ + 8x² + 5

The degree of the polynomial is the highest exponent. The leading coefficient is the number in front of the term with the highest exponent because when the polynomial is in standard form, the term with the highest exponent is goes first (that is, it's the leading term). The constant, which usually goes last, doesn't have any variable.

The only polynomial that fits the conditions is Choice (3).

Note: This is almost EXACTLY the question that was asked in January for question number 6.

Choice (1) has a constant of -5.

Choices (2) and (4) have degree 4 and a leading coefficient of 3.

7. Given the sequence 128,64,32,..., which formula could be used to find the nth term of this sequence?
(1) a_n = 128(-2)^{n - 1}
(2) a_n = 128(1/2)^{n - 1}
(3) a_n = 128 - 2(n - 1)
(4) a_n = 128 + (1/2)(n - 1)

Answer: (2) a_n = 128(1/2)^{n - 1}

Each term is one half of the previous term. The Common Ratio is 1/2. This is a geometric sequence. Eliminate Choices (3) and (4).

In Choice (1), the sign would be changing, and every other term would be four times greater than the term two before it. Eliminate Choice (1).

Choice (2) is the Correct choice.

8. When a bicyclist increases the pressure on the brakes, the speed of the bicycle decreases. This relationship can best be described as a

(1) negative correlation and causal relationship
(2) negative correlation and non-causal relationship
(3) positive correlation and causal relationship
(4) positive correlation and non-causal relationship

Answer: (1) negative correlation and causal relationship

The brakes slow the bike. The more pressure on the brakes, the slower the speed of the bike.

The correlation is causal and negative.

Note that the word is "causal" and NOT "casual". If you misread this, you may be somewhat confused because that doesn't mean anything.

9. A garden club plans to plant 40 flowering plants this year. They will only purchase daffodils that cost $4 per plant and tulips that cost $5 per plant. All prices include tax. The club has $170 to spend on plants. Which equation could be used to find the number of daffodil plants, d, the club purchases?

(1) 4d + 5(170 − d) = 40
(2) 4d + 5(40 − d) = 170
(3) 5d + 4(170 − d) = 40
(4) 5d + 4(40 − d) = 170

Answer: (2) 4d + 5(40 − d) = 170

The total is $170, so eliminate Choices (1) and (3).

Daffodils cost $4, so the cost of all the daffodils bought will be 4d. Eliminate Choice (1).

The correct choice is Choice (2).

A system of equations could be 4d + 5t = 170 and d + t = 40, so t = 40 - d, which could be substituted into the first equation.

10. A function is defined as h(x) = x² − 3x + 1. What is the value of h(−1)?

(1) 1
(2) 2
(3) 3
(4) 5

Answer: (4) 5

Substitute and evaluate. Make sure you use parentheses around the -1.

h(-1) = (-1)² − 3(-1) + 1 = 1 + 3 + 1 = 5

The correction substitution is Choice (4).

11. The zeros of p(x) = x(3x + 2)(x − 5) are BR>
(1) -2/3 and 5, only
(2) 2/3 and -5, only
(3) -2/3, 0, 5
(4) 2/3, 0, -5

Answer: (3) -2/3, 0, 5

There are three distinct linear factors, so there will be three distinct zeroes. Eliminate Choices (1) and (2).

You could graph the function and look at the table of values (or the graph for that matter).

If x - 5 = 0, then x = 5, this is in Choice (3).

If 3x + 2 = 0, then 3x = -2 and x = -2/3, which is also Choice (3).

The correct choice is Choice (3).

12. If f(x) = 1.25^x and g(x) = 3x + 10, what is the smallest positive integer of x for which f(x) > g(x)?

(1) 18
(2) 19
(3) 67
(4) 69

Answer: (2) 19

You could graph the function and look at the table of values.

f(18) = 55.511..., g(18) = 64
f(19) = 69.388..., g(19) = 67

Choice (2) is the correct answer.

Note that the values of g(19) and f(19) are Choices (3) and (4), respectively.

13. What is an equation of the line that passes through the points (2,5) and (−2,−1)?

(1) y - 5 = 2/3 (x - 2)
(2) y - 5 = 3/2 (x - 2)
(3) y - 2 = 2/3 (x - 5)
(4) y - 2 = 3/2 (x - 5)

Answer: (2) y - 5 = 3/2 (x - 2)

The formula is y - y₁ = m(x - x₁) -- or sub 0, if you prefer. Either way, for the point (2,5), x = 2 and y = 5. Eliminate Choices (3) and (4).

Calculate the slope: m = (-1 - 5) / (-2 - 2) = (-6)/(-4) = 3/2. So Choice (2) is the correct answer.

14. At a local high school, students were asked to name the sport they like to watch the most. The results are summarized in the table below.

Approximately what percentage of female preferred to watch basketball?

(1) 35
(2) 40
(3) 53
(4) 57

Answer: (4) 57

You only care about the "Female" row of the table, but you need to find the total number of Females who responded.

THere were 40 who liked to watch basketball out of (20 + 40 + 10) = 70, and 40/70 = 0.5714..., which is about 57%.

The correct choice is Choice (4).

15. An equation that yields the same solutions as x² − 10x − 24 = 0 is

(1) (x + 5)² = 1
(2) (x - 5)² = 1
(3) (x + 5)² = 49
(4) (x - 5)² = 49

Answer: (4) (x - 5)² = 49

Start by Completing the Square.

x² − 10x − 24 = 0
x² − 10x + 25 − 24 = 25
x² − 10x + 25 = 49
(x - 5)² = 49

The correct choice is Choice (4).

16. The function f(x) is shifted three units right and four units up. The result of this transformation is

(1) f(x − 3) + 4
(2) f(x + 3) + 4
(3) f(x + 4) - 3
(4) f(x - 4) - 3

Answer: (1) f(x − 3) + 4

If the function is shifted up four units, then four will be added to the function OUTSIDE of the parentheses. Eliminate Choices (3) and (4).

When a function is shifted to the right three units, then three will be subtracted INSIDE the parentheses.

The correct choice is Choice (1).

17. Hana was asked to solve a quadratic equation. Her first step is shown below. x² − 8x = 3
Step 1: x² − 8x + 16 = 3 + 16
The property that Hana used is the

(1) distributive property
(2) commutative property
(3) additive inverse property
(4) addition property of equality

Answer: (4) addition property of equality

The same amount is added to each side of the equal sign. If x = y then x + n = y + n.

The correct choice is Choice (4).

18. A data set is given below: 28 28 28 28 32 32 34 34 40 42

What is the value of the upper quartile of this data set?

(1) 28
(2) 32
(3) 34
(4) 42

Answer: (3) 34

There are 10 pieces of data. The upper quartile is the median of the upper half of the data (the top five numbers). That number is 34.

The upper quartile (Q3) is 34, which is Choice (3).

19. When −3x² + 7x − 1 is subtracted from 2x² − 3x + 10, the result is

(1) 5x² + 4x + 9
(2) 5x² - 10x + 11
(3) -5x² + 4x + 9
(4) -5x² + 10x - 11

Answer: (2) 5x² - 10x + 11

When subtracting polynomials, what follows the "from" goes on top:

2x² − 3x + 10
−3x² + 7x − 1
5x² - 10x + 11

The correct choice is Choice (2).

20. The value of a home in Buffalo can be modeled by the function V(t) = 96,949(1.0448)^t , where V(t) is the value of the house after t years. What is the percent of increase in the value of the home each year?

(1) 1.0448%
(2) 0.0448%
(3) 0.448%
(4) 4.48%

Answer: (4) 4.48%

The 1.0448 means that the value is increasing by a factor of 0.0448, which is 4.48%

The correct choice is Choice (4).

21. A rod is an old English measure of distance that is equivalent to 5.5 yards. How many inches are 2.5 rods? [1 yard = 3 feet]

(1) 66
(2) 165
(3) 198
(4) 495

Answer: (4) -51.9

There are 5.5 yards in one rod. There are 3 feet in one yard. There are 12 inches in one foot.

Therefore, there are 2.5 * 5.5 * 3 * 12 = 495 inches in 2.5 rods. This is Choice (4).

Choice (1) is the answer you would get if you multiply 5.5 * 12 and forgot the 3 and the 2.5.

Choice (2) is the answer you would get if you forgot to multiply by 3.

Choice (3) is the number of inches in one rod.

I'm surprised none of the incorrect choices had the number of feet, which is 41.25.

22. When solving the equation 2x² − 3x − 6 = 0 using the quadratic formula, the solutions are

(1) (3 + √(57)) / 4
(2) (3 + √(39)) / 4
(3) (-3 + √(57)) / 4
(4) (-3 + √(39)) / 4

Answer: (1) (3 + √(57)) / 4

Using the quadratic formula in this case, a = 2, b = -3, and c = -6. This means that (-b) = (-(-3)) = 3. Eliminate Choices (3) and (4).

The discriminat has the formula b² - 4ac. Remember to use parentheses!

Thus, (-3)² - 4(2)(-6) = 9 - (-48) = 57. Eliminate Choices (2) and (4).

The correct choice is Choice (1).

23. For which function is the axis of symmetry x = −4?

(1) f(x) = -x² - 4x - 1
(2) g(x) = -x² + 8x + 5
(3) h(x) = x² + 8x + 3
(4) k(x) = x² + x - 4

Answer: (3) (x - 3)² = 33

You can graph each of these parabolas and check. Or you can use the formula x = -b/(2a) to see which is x = -4.

In Choice (1), x = -(-4) / (2(-1) = -2. Eliminate Choice (1).

In Choice (2), x = -(8) / (2(-1) = 4. Eliminate Choice (2).

In Choice (3), x = -(8) / (2(1) = 4. This is the correct answer.

In Choice (4), x = -(1) / (2(1) = -1/2. Eliminate Choice (4).

24.In simplest radical form, the product of 2√(6) and 5√(3) is

(1) 10
(2) 21
(3) 30√(2)
(4) 10√(18)

Answer: (3) 30√(2)

Multiply the terms and then simplify the radical.

(2√(6))(5√(3))
= (10)√( (6)(3) )
= (10)√( (2)(3)(3) )
= (3)(10)√(2)
= 30√(2)

This is Choice (3).

Note that Choice (4) is the same amount, but it isn't simplified. The radicand 18 can be factored into (9)(2) and (9) is a perfect square.

More to come. Comments and questions welcome.

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My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra Problems of the Day (Algebra Regents, January 2026 Parts III and IV)

This exam was adminstered in January 2026.

January 2026 Algebra, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

31. A rocket was launched from the ground into the air at an initial velocity of 80 feet per second.

The path of the rocket can be modeled by h(t) = -16t² + 80t, where t represents the time after the rocket has been launched, and h(t) represents the height of the rocket.

Sketch the function on the set of axes below.

State how many seconds it will take for the rocket to reach its maximum height.

State the maximum height, in feet, of the rocket.

Answer:

Use your graphing calculator to find the points that you need to plot before t = 0 and when h(t) = 0.

You'll get the following points: (0,0), (1,64), (2,96), (3,96), (4,65), (5,0).

If the zeros of the rocket's path are 0 and 5, then the highest point occurs when t = 2.5. It should be "obvious" that the rocket does not just hang in the air at 96 feet for an entire second.

You can adjust the calculator to show increases of .5 for t (or x, on the calculator). When t = 2.5, h(t) = 100.

Your graph should look like this:

Do not draw arrows below the ground.

The rocket reached its maximum height at 2.5 seconds. The rocket reached a maximum height of 100 feet.

32. Use the quadratic formula to solve 2x² -4x - 3 = 0, and express the answer in simplest radical form.

Answer:

Use the quadratic formula with a = 2, b = -4, and c = -3. Make sure you use parentheses around the negative numbers!

( -(-4) + √( (-4)² - 4(2)(-3) ) ) / ( 2(2))
( 4 + √( 16 - (-24) ) ) / 4
( 4 + √(40) ) / 4
( 4 + √( (4)(10) ) ) / 4
( 4 + 2√(10) ) / 4
1 + 1/2 √(10)

Either of the last two lines are acceptable. So long as the radical is simplified, it doesn't matter if the fraction is.

33. The table below shows the ages of drivers and the annual cost of their car insurance.

Write the linear regression equation for this set of data. Round all values to the nearest hundredth.

State the correlation coefficient of this line of best fit, to the nearest hundredth.

State what this correlation coefficient indicates about the linear fit of the data set.

Answer:

Put the data into Lists L₁ and L₂. Make sure that "DiagnosticOn" has been executed. (It's in the Catalogue.) Run a Linear Regression.

COPY THE ENTIRE WINDOW OF DATA ONTO YOUR TEST PAPER. The reason for this is if you entered any piece of information incorrectly, all of the following answers are likely to be incorrect. However, if you show the data that you were given by your calculator, you can still earn three of the four points.

The equation for the linear equation will be y = -56.97x + 2352.22

The correlation coefficient is r = -0.98.

The correlation coefficient indicates a strong negative correlation.

There isn't much to say about this. Either you know how to enter the data and run the linear regression, or you do not.

34. Solve the following system of inequalities graphically.

Label the solution set S

2y < x + 6
2x + y > 3

Is the point (0, 3) in the solution set? Explain your answer.

Answer:

Rewrite the inequalities, solving for y in terms of x and graph.

Remember that < means a solid line that gets shading underneath, and > means a broken line that gets shaded above.

2y < x + 6
y < 1/2 x + 3

2x + y > 3
y > -2x + 3

Your graph should look like this:

(0,3) is not in the solution because it's on the broken line, which is not part of the solution.

June 2025 Algebra, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. Acme Athletics purchases shoes from a supply company. In January the store bought 30 pairs of running shoes and 10 pairs of basketball shoes for $3700. In March they bought 15 pairs of running shoes and 20 pairs of basketball shoes for $3575. The supply company kept their prices constant.

If x represents the cost of one pair of running shoes and y represents the cost of one pair of basketball shoes, write a system of equations that models this situation.

Jacob says that a pair of running shoes costs the store $80 each, and a pair of basketball shoes costs the store $130 each. Is he correct? Justify your answer.

Solve your system of equations algebraically to find the exact cost, in dollars, of one pair of running shoes and the exact cost, in dollars, of one pair of basketball shoes.

Answer:

The second sentence give us 30x + 10y = 3700. The third sentence gives us 15x + 20y = 3575. (The fourth sentence tells us to use x and y, so use them!)

Check Jacbo's answer by substituting his values into both equations.

30(80) + 10(130) = 2400 + 1300 = 3700 (check)
15(80) + 20(130) = 1200 + 2600 = 3800, not 3575.
Jacob is incorrect.

Solve the system of eqautions.

30x + 10y = 3700
15x + 20y = 3575

60x + 20y = 7400
15x + 20y = 3575
45x = 3825
x = 85

Solve for y:

30(85) + 10y = 3700
2550 + 10y = 3700
10y = 1150
y = 115

End of Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
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If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra Problems of the Day (Algebra Regents, January 2026 Part II)

This exam was adminstered in January 2026.

January 2026 Algebra, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. Solve the equation for x:

14x = 3(1 + 2x) - 4x

Answer:

Use the Distributive property, Combine like terms, and then Inverse operations:

14x = 3(1 + 2x) - 4x
14x = 3 + 6x - 4x
14x = 3 + 2x
12x = 3
x = 3/12 = 1/4

26. Graph f(x) = 3(2)^x over the interval -1 < x < 2.

Answer:

Put the equation into your graphing calculator and plot the points that appear in the Table of Values. Start at x = -1 and end at x = 2. Do NOT go beyond. Do NOT draw arrows. You MUST have TWO endpoints on your graph.

Your graph should look like this:

27. Determine the product of (2x + 3) and (-6x² + 5x - 1). Express the product in standard form.

Answer:

Multiply and combine like terms.

(2x + 3) (-6x² + 5x - 1)
(2x)(-6x² + 5x - 1) + (3)(-6x² + 5x - 1)
(-12x³ + 10x² - 2x) + (-18x² + 15x - 3)
-12x³ - 8x² + 13x - 3

28. A student’s test scores for the semester are listed below.

83, 87, 90, 94, 94, 93, 95, 70, 72, 83, 85, 88, 98

Construct a box plot for this data set, using the number line below.

Answer:

You can use a graphing calculator to give you a Five Number Summary of this data by putting it into a List.

Or you can find the Five-Number Summary yourself after you put the data into order.

70, 72, 83, 83, 85, 87, 88, 90, 93, 94, 94, 95, 98.

The minimum is 70, and the maximum is 98. The median of 13 pieces of data will be the 7th entry, which is 88.

There are six numbers less than 88. The two numbers in the middle are both 83, so Q1 is 83.

There are six numbers greater than 88. The two numbers in the middle are both 94, so Q1 is 94.

Plot these five points. Draw the box around the three middle points and draw the whiskers.

You should get something like this.

29. Write an equation, in slope-intercept form, of a line that passes through the point (6, 3) and has a slope of 2/3.

Answer:
You can start with point-slope form and rewrite the equation in slope-intercept form, or you can start with slope-intercept form, and solve for the y-intercept.

Method 1:

y - 3 = 2/3 (x - 6)
y - 3 = 2/3 x - 4
y = 2/3 x - 1

Method 2:

y = mx + b
3 = (2/3)(6) + b
3 = 4 + b
-1 = b
y = 2/3 x - 1

30. Abby has $20 to spend at a community festival. She uses $8.50 to purchase food coupons for popcorn, a hot dog, and a soda.

She can buy individual ride tickets for $2.25 each. Determine algebraically the maximum number of ride tickets Abby can buy.

Answer:
Write an inequality and then solve it.

2.25x + 8.50 < 20

2.25x < 11.50

x < 5.11...

She can buy five ride tickets.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra Problems of the Day (Algebra Regents, January 2026 Part I)

This exam was adminstered in January 2026.

More Regents problems.

January 2026 Algebra Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


1. A parabola is graphed on the set of axes below.
What are the equation of the axis of symmetry and the coordinates of the vertex of this parabola?

(1) x = 3 and (3, -4)
(2) y = 3 and (3, -4)
(3) x = -4 and (-4, 3)
(4) y = -4 and (-4, 3)

Answer: (1) x = 3 and (3, -4)

The axis of symmentry is a vertical line that goes through the vertex of the parabola with an equation x = a, where a is the x-coordinate of the vertex.

The vertex is (3, -4) and the axis of symmetry is x = 3.

The correct answer is Choice (1).

2. The product of &sqrt;(25) and &sqrt;(2) will result in

(1) an irrational number
(2) a rational number
(3) a natural number
(4) an integer

Answer: (1) an irrational number

The product of &sqrt;(25) and &sqrt;(2) is &sqrt;(50), which is irrational because 50 is not a perfect square.

The product MUST be either Rational or Irrational, so Choices (3) and (4) could be eliminated because all natural numbers are also integers and all integers are rational numbers.

The correct answer is Choice (1).

3. When f(x) = |4x + 2| and g(x) = 3x + 5 are graphed on the same set of axes, for which value of x is f(x) = g(x)?

(1) 1
(2) 2
(3) 3
(4) 14

Answer: (3) 3

You can plug each choice into each function using the calculator, or just graph them. Or you can solve it.

4x + 2 = 3x + 5
x = 3

4x + 2 = -1(3x + 5)
4x + 2 = -3x - 5
7x = -7
x = -1

The two functions intersect at x = -1 and x = 3.

The correct Choice is Choice (3).

4. The expression x² - 26x - 120 is equivalent to

(1) (x + 4)(x - 30)
(2) (x - 4)(x + 30)
(3) (x - 20)(x + 6)
(4) (x + 20)(x - 6)

Answer: (1) (x + 4)(x - 30)

The factors of the constant term, -120, have to add up to coefficient of the middle term, which is -26. This means that there will be one positive and one negative term and that the negative term must be bigger.

Choice (1) is the correct answer.

Choice (2) gives a middle term of + 26x. Choice (3) gives a middle term of -14x. Choice (4) gives a middle term of -14x.

5. The expression 3 - 2&sqrt;(5) + 6&sqrt;(5) is equivalent to

(1) 7&sqrt;(5)
(2) 7&sqrt;(10)
(3) 3 + 4&sqrt;(5)
(4) 3 + 4&sqrt;(10)

Answer: (3) 3 + 4&sqrt;(5)

You can only combine like terms. Imagine this was 3 - 2x + 6x. That would simply to 3 + 4x.

Now replace the x with &sqrt;(5). The radicand (the number under the radical) doesn't change in the same way that exponents don't change when you add terms with exponents.

The correct answer is (3).

6. Students were asked to write a polynomial given the following conditions:
• the degree of the expression is 3
• the leading coefficient is 2
• the constant term is -6
Which expression satisfies all three conditions?

(1) 4x - 6 + 3x²
(2) 3x² - 6x + 4
(3) 4 - 6x + 2x³
(4) 4x² + 2x³ - 6

Answer: (4) 4x² + 2x³ - 6

The degree of the polynomial is the highest exponent. The leading coefficient is the number in front of the term with the highest exponent because when the polynomial is in standard form, the term with the highest exponent is goes first (that is, it's the leading term). The constant, which usually goes last, doesn't have any variable.

The only polynomial that fits the conditions is Choice (4).

Choices (1) and (2) have degree 2. Choice (3) has a constant of 4.

7. Which graph below represents a function?

Answer: (1) [See Image]

A graph is a function so long as no x value has two corresponding y values. In other words, it will pass the Vertical Line Test in that no vertical line can be drawn that goes through two points.

Choice (1) has no points that line up vertically. This is the Correct choice.

Choice (2) has two pairs of vertical points. Eliminate Choice (2).

Choice (3) has two values when x = 4. Eliminate Choice (3).

Choice (4) has portions of three vertical lines. It fails the veritcal line test spectacularly. Eliminate Choice (4).

8. The following function models the value of a diamond ring, in dollars, t years after it is purchased:

v(t) = 500(1.08)^t

What was the original price of the ring, in dollars?

(1) $108
(2) $460
(3) $500
(4) $540

Answer: (3) $500

The original price happens when t=0. At t=0, v(t) = 500. This is Choice (3).
9. The formula for the surface area of a cylinder can be expressed as S = 2πr² + 2πrh, where r is the radius and h is the height of the cylinder. What is the height, h, expressed in terms of S, π, and r?

(1) h = (S - 2πr²) / (2πrh)
(2) h = S - r
(3) h = (2πr² - S) / (2πrh)
(4) h = r - S

Answer: (1) h = (S - 2πr²) / (2πrh)

Use inverse operations:

S = 2πr² + 2πrh

S - 2πr² = 2πrh

(S - 2πr²) / (2πr) = h

The correct choice is Choice (1).

10. When solving the following system of equations algebraically, Mason used the substitution method. 3x - y = 10
2x + 5y = 1
Which equation could he have used?

(1) 2(3x - 10) + 5x = 1
(2) 2(-3x + 10) + 5x = 1
(3) 2x + 5(3x - 10) = 1
(4) 2x + 5(-3x + 10) = 1

Answer: (3) 2x + 5(3x - 10) = 1

Changing 5y to 5x is just DUMB. Eliminate Choices (1) and (2).

Rewrite the first question to solve for y and then substitute for y in the second equation.

3x - y = 10
3x - 10 = y

The correction substitution is Choice (3).

11. Which graph represents the solution to the inequality 4 + 3x > 9 - 7x?

Answer: (3) [See Image]

The quick way to get this answer is to check x = 2 to see if the two expressions are equal. Then check x = 0:

If x = 2, then 4 + 6 = 10 and 9 - 14 = -5. Those aren't equal so eliminate Choice (1) and (2).

If x = 0, then 4 + 0 > 9 - 0 is NOT a true statement. Therefore x = 0 is NOT in the solution to the inequality. Eliminate Choice (4).

The correct choice is Choice (3).

Doing the work:

4 + 3x > 9 - 7x
10x > 5
x > 1/2

Choice (3) shows the graph of x > 1/2.

12. When solving the equation 3(2x + 5) - 8 = 7x + 10, the first step could be 3(2x + 5) = 7x + 18. Which property justifies this step?

(1) addition property of equality
(2) commutative property of addition
(3) multiplication property of equality
(4) distributive property of multiplication over addition

Answer: (1) addition property of equality

In the first step, 8 was added to both sides of the equation. This is Choice (1).

13. Which table of values best models an exponential decay function?

Answer: (2) [See Image]

A decay function will decrease with a constant ratio. It will NOT have a constant rate of change.

Choice (1) decreases with a constant rate of 3. Eliminate Choice (1).

Choice (2) decreases by 20 then 18, then 16, then 15.
-20/200 = -.1, -18/180 = -.1, etc. The function is decreasing by 10% (and rounded to the nearest integer). This is the correct choiuce.

Choice (3) is increasing, so it isn't a decay function. Eliminate Choice (3).

Choice (4) decreases and then increases again, so it isn't a decay function. Eliminate Choice (4).

The correct choice is Choice (2).

14. If f(x) = √(x + 1) + 5, then what is the value of f(3)?

(1) 9
(2) 7
(3) 3
(4) 10

Answer: (2) 7

Because careful entering this into a calculator that you don't accidentally evaluate √(x + 1 + 5).

√(3 + 1) + 5 = √(4) + 5 = 2 + 5 = 7

The correct choice is Choice (2).

If you used √(x + 1 + 5), you would've gotten 3 as an answer and wouldn't have realized your error.

15. Isabella wants to shift the graph of the function f(x) = (x + 5)² - 2 left 3 units. Which function represents the shifted graph?

(1) g(x) = (x + 2)² - 2
(2) g(x) = (x + 8)² - 2
(3) g(x) = (x + 5)² - 5
(4) g(x) = (x + 5)² + 1

Answer: (2) g(x) = (x + 8)² - 2

The vertex is at (-5, -2). If it is shifted left 3 spaces, the vertex of the new function will be (-8,-2). The y-coordinate will not change.

This means that the new function will be g(x) = (x + 8)² - 2.

The correct choice is Choice (2).

16. What are the zeros of f(x) = x(x² - 36)?

(1) 0, only
(2) 6, only
(3) 6 and -6, only
(4) 0, 6, and -6

Answer: (4) 0, 6, and -6

If you factor x² - 36 into (x + 6)(x - 6), then you can see that the function has three terms with three distinct zeros, which are 0, -6 and 6.

Otherwise, you can graph the function, or you can substitute each of the three values to see what works.

The correct choice is Choice (4).

17. The point (x, -6) lies on the graph of a parabola whose equation is y = -x² - x + 6. The value of x can be

(1) -3 or 2
(2) -4 or 3
(3) 3, only
(4) -4, only

Answer: (2) -4 or 3

Graph the equation, or substitute each value until you get a correct answer.

Or you can solve for x:

x² - x + 6 = -6
x² - x + 12 = 0
(x - 4)(x + 3) = 0
x - 4 = 0 or x + 3 = 0
x = 4 or x = -3

The correct choice is Choice (2).

18. The two-way frequency table below is a summary of concession stand sales for a football game.

Of the people making a purchase at the concession stand, what is the relative frequency of them buying pizza and a water?

(1) 0.58
(2) 0.35
(3) 0.455
(4) 0.145

Answer: (4) 0.145

There were 58 people who bought water and pizza out of 400 people who made purchases.

To get the relative frequency, divide (but DO NOT make it a percentage):

58/400 = 0.145

The correct choice is Choice (4).

19. When Theodore was driving in Canada, his speed was 104 kilometers per hour. Theodore was asked to convert his metric speed to a different rate, using the following conversion:

Assuming he did all the work correctly, what would the units be for Theodore’s rate?

(1) feet per second
(2) feet per minute
(3) seconds per foot
(4) minutes per foot

Answer: (1) feet per second

Units can be multiplied and divided just like any other factor.

Cancel out any units that appear once in the numerator (top) and once in the denominator (bottom).

You'll get this:

That leaves feet per second.

The correct choice is Choice (1).

20. Which expression is equivalent to (-2x²)³?

(1) -2x⁵
(2) -2x⁶
(3) -8x⁵
(4) -8x⁶

Answer: (4) -8x⁶

(-2x²)³ is equivalent to (-2x²)(-2x²)(-2x²).

(-2)(-2)(-2) = -8, and (x²)(x²)(x²) = (x⁶)

The correct choice is Choice (4).

21. The table below shows the amount of a radiactive substance that remained for selected years.

To the nearest tenth, the average rate of change, in grams per year, from 2000 to 2014 is

(1) 39.1
(2) 51.8
(3) -39.1
(4) -51.9

Answer: (4) -51.9

Ignore every column except 2000 and 2014. Find the rate of change for those two points.

First off, the rate of change is negative, so Eliminate Choices (1) and (2).

Calculate (25 - 750) / (2014 - 2000) = -725/14, or about 51.8.

The correct choice is Choice (4).

22. When 2x² -3x + 4 is subtracted from x² + 2x - 5, the result is

(1) x² - 5x + 9
(2) x² - x + 1
(3) -x² + 5x - 9
(4) -x² - x - 1

Answer: (3) $500

When subtracting the expression AFTER the "from" goes first.

x² + 2x - 5
2x² -3x + 4
-x² + 5x - 9

The correct choice is Choice (3).

23. Which equation has the same solution as x² - 6x = 24?

(1) (x - 3)² = 24
(2) (x - 6)² = 24
(3) (x - 3)² = 33
(4) (x - 6)² = 60

Answer: (3) (x - 3)² = 33

When Completing the Square, the number in the parentheses will be half the amount of the coefficient of the x term in the original equation. Eliminate Choices (2) and (4).

It should be "obvious" that Choice (1) is incorrect, but just in case, let's work it out:

Half of -6 is -3, and -3 squared is +9, so 9 must be added to both sides of the equation.

x² - 6x = 24

x² - 6x + 9 = 24 + 9

(x - 3)² = 33

The correct choice is Choice (3).

24. In a sequence, the first term is -2 and the common ratio is -3. The fourth term in this sequence is

(1) -162
(2) -11
(3) 24
(4) 54

Answer: (4) 54

You only need the fourth term, so skip the formulas and just work out the first four terms. Start with -2 and multiply each term by -3 to get the next term.

-2, 6, -18, 54.

The fourth term is 54. (The fifth term would be -162.)

The correct choice is Choice (4).

More to come. Comments and questions welcome.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!