Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


21. Which equation is represented by the graph below?

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

Answer: 4) 2y = -4x - 10

You can pick a couple of points (especially the y-intercept) from the graph and try them in the equations. Only one equation can have two points as solutions because two points define a line.

2y + x = 10, 2(-5) + 0 =/= 10. Eliminate Choice (1).

y - 2x = -5, -5 - 2(0) = -5. Try a second point.
(-4, 3): 3 - 2(-4) =/= -5. Eliminate Choice (2).

-2y = 10x - 4, -2(-5) =/= 10(0) - 4. Eliminate Choice (3).

2y = -4x - 10, 2(-5) = -4(0) - 10.
(-4, 3): 2(3) ?= -4(-4) - 10
6 = 16 - 10 = 6. Choice (4) is the answer.

The alternative is to find the equation of the line and see which of the choices it resembles.

The slope is -2 rise and 1 run, or m = -2. The y-intercept is -5, so the equation for this line is y = -2x - 5, which is 2x + y = -5 in Standard form, which is NOT the same as (2).

If you multiply the equation by 2, you get 4x + 2y = -10, which then becomes 2y = -4x - 10.

22. Which coordinates represent a point in the solution set of the system of inequalities shown below?

y < 1/2x + 13
4x + 2y > 3

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

Answer: 4) (2, -2)

Pick either of the two inequalities and try the points out.

4(-4) + 2(1) = -13, which is not greater than 3. Eliminate Choice (1).

4(-2) + 2(2) = -4, which is not greater than 3. Eliminate Choice (2).

4(1) + 2(-4) = -4, which is not greater than 3. Eliminate Choice (3).

4(2) + 2(-2) = 4 > 3.
3 ?< 1/2 (2) + 13
3 < 14. Choice (4) is the answer.

23.The length of one side of a square is 13 feet. What is the length, to the nearest foot, of a diagonal of the square?

1) 13
2) 18
3) 19
4) 26

Answer: 2) 18

If you don't know the shortcurt, use the Pythagorean Theorem. However, at the very least, you know that you can eliminate 13, because it has to be longer, and 26, because 13-13-26 does NOT make a triangle.

a² + b² = c²
13² + 13² = c²
169 + 169 = c²
338 = c²
c = 18.38...

The diagonal, to the nearest foot, is 18.

The shortcut is realizing that two sides of a square and a diagonal make an isosceles right triangle where the hypotenuse is the length of one leg times the square root of two.

13 * sqrt(2) = 18.38...

24. In triangle ABC, m∠C = 90. If AB = 5 and AC = 4, which statement is not true?

1) cos A = 4/5
2) tan A = 3/4
3) sin B = 4/5
4) tan B = 5/3

Answer: 4) tan B = 5/3

If the right triangle has a leg of 4 and hypotenuse of 5, then the other leg is 3. We should show this with Pythagorean Theorem, but really, by this point, you should just KNOW that 3-4-5 is the simplest, smallest Pythagorean Triple.

The rules for sin, cos and tan are: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

Cos A = adj / hyp = 4/5

Tan A = opp / adj = 3/4

Sin B = opp B / hyp = 4/5, which is the same as Cos A.

Tan B = opp B / adj B = 4/3, not 5/3. The hypotenuse is 5, but it is NOT used in tangents.

25. If n is an odd integer, which equation can be used to find three consecutive odd integers whose sum is -3?

1) n + (n + 1) + (n + 3) = -3
2) n + (n + 1) + (n + 2) = -3
3) n + (n + 2) + (n + 4) = -3
4) n + (n + 2) + (n + 4) = -3

Answer: 3) n + (n + 2) + (n + 4) = -3

If n is an odd number, then the next odd number is n + 2 and the next one after that is n + 4. The sum of them equals -3.

More to come. Comments and questions welcome.

More Regents problems.