Algebra Problems of the Day (Algebra Regents, January 2025 Part III)

This exam was adminstered in January 2025.

January 2025 Algebra, Part III

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

31. Alex had $1.70 in nickels and dimes on his desk. There were 25 coins in all. Write a system of equations that could be used to determine both the number of nickels, n, and the number of dimes, d, that Alex had.

Use your system of equations to algebraically determine both the number of nickels and the number of dimes that he had.

Answer:

Write the system of equations for the number of nickels and dimes and the value of the nickels and dimes. Then solve the system.

.05n + .10d = 1.70
n + d = 25

.5n + d = 17.0
n + d = 25

.5n = 8
n = 16

16 + d = 25
d = 9

Alex has 16 nickels and 9 dimes.

32. The table below shows the average heart rate, x, and Calories burned, y, for seven men on an Olympic rowing team during a one-hour workout class.

Write the linear regression equation that models these data, rounding all values to the nearest tenth.

State the correlation coefficient, rounded to the nearest tenth.

State what the correlation coefficient suggests about the linear fit of these data.

Answer:

Put the data into the L1 and L2 in your graphing calculator and run a linear regressions.

Rounding to the nearest tenth, you'll get y = 9.1x - 527.6. Remember to write the equation. Put just write a = 9.1 and b = 527.6.

If the correlation coefficient didn't appear on your screen, go to the Catalogue and run "DiagnosticOn" and do the regression a second time.

To the nearest tenth, r = 0.9.

This suggests that there is a strong positive correlation between average heart rate and calories burned.

33. Using the quadratic formula, solve x² + 4x - 3 = 0.
Express your solution in simplest radical form.

Answer:

The equation is in the back of the booklet. (Lucky you -- back in my century, we had to memorize it!)

x = (-b + √(b² - 4ac)) / (2a)

x = (-4 + √(4² - 4(1)(-3)) / (2(1))

x = (-4 + √(16 + 12)) / 2

x = (-4 + √(28)) / 2

x = (-4 + (√(4)*√(7)) / 2

x = (-4 + 2√(7)) / 2

x = -2 + √(7)

The next to last line would have been an acceptable answer.

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

y = x² - 7x + 12
y = 2x - 6

Answer:

Set the two expressions equal to each other and then solve the quadratic equation. Then solve for y for each value of x.

x² - 7x + 12 = 2x - 6
x² - 9x + 18 = 0
(x - 3)(x - 6) = 0
x - 3 = 0 or x - 6 = 0
x = 3 or x = 6

If x = 3: y = 2(3) - 6 = 0
If x = 6: y = 2(6) - 6 = 6.

End of Part III

How did you do?

Questions, comments and corrections welcome.

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