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.


11. If five times a number is less than 55, what is the greatest possible integer value of the number?

1) 12
2) 11
3) 10
4) 9

Answer: 3) 10

If five times a number is less than 55 than one times the number is less than 11. But it is NOT 11.

12. The line represented by the equation 2y - 3x = 4 has a slope of

1) -3/2
2) 2
3) 3
4) 3/2

Answer: 4) 3/2

Don't get fooled by the y term being written first. It isn't written in Standard Form, in case you want to use a formula that required standard form.

You can rewrite the equation, solving for y:

2y - 3x = 4
2y = 3x + 4
y = 3/2 x + 2

Or you can switch around the terms and use the formula:

2y - 3x = 4
-3x + 2y = 4
Ax + By = C
Slope equals -A / B = -(-3) / 2 = 3/2

13. What is the solution set of the system of equations x + y = 5 and y = x² - 25?

1) {(0,5), (11, -6)}
2) {(5,0), (-6,11)}
3) {(-5,0), (6,11)}
4) {(-5,10), (6,-1)}

Answer: 2) {(5,0), (-6,11)}

You can solve the systems of equations, or you can check the solutions to see which sets have two points that satisfy both equations.

If you checking the solutions, start with x + y = 5 to see if anything can be eliminated, because it's the easiest to deal with

0 + 5 = 5, 11 + -6 = 5

5 + 0 = 5, -6 + 11 = 5

-5 + 0 =/= 5, Eliminate Choice (3)

-5 + 10 = 5, 6 + -1 = 5

Next try the other equation. Remember, you can skip Choice (3):

0² - 25 =/= 5, Eliminate Choice (1)

5² - 25 = 0, (-6)² - 25 = 36 - 25 = 11. Check. Choice (2) is the solution.

(-5)² - 25 =/= 10. Eliminate Choice (4).

14. What is the vertex of the parabola represented by the equation y = -2x² + 24x - 100?

1) x = -6
2) x = 6
3) (6, -28)
4) (-6, -316)

Answer: 3) (6, -28)

The vertex of a parabola is a point, not a line. So eliminate Choices (1) and (2). One of those might be the Axis of Symmetry which will go through the vertex, but neither is the vertex.

The Axis of Symmetry is x = -b / (2a) = -24 / (2(-2)) = 6

The y-coordinate of the vertex is y = -2(6)² + 24(6) - 100 = -28. So (6, -28)

You also could have found this by putting the equation in your graphing calcuator and looking at the table of values to find the turning point. Note that since (6, -28) is a MAXIMUM, you wouldn't see anything on your screen unless you changed the scale shown in the window.

15. If k = am + 3mx, the value of m in terms of a, k, and x can be expressed as

1) k / (a + 3x)
2) (k - 3mx) / a
3) (k - am) / 3x
4) (k - a) / 3x

Answer: 1) k / (a + 3x)

If you are solving for m in terms of a, k and x, then you will NOT have m in the solution. You can eliminate (2) and (3) immediately.

Solve for m by using Inverse Operations to isolate the variable:

k = am + 3mx
k = m(a + 3x)
k / (a + 3x) = m

More to come. Comments and questions welcome.

More Regents problems.