Question

The relationship between two numbers is described below, where x represents the first number and y represents the second number.

The square of the first number is equal to the sum of the second number and 16. The difference of 4 times the second number and 1 is equal to the first number multiplied by 7.

Select the equations that form the system that models this situation. Then, select the solution(s) of the system.

Answer 1

See below.

Explanation:

x^2 = y + 16

4y - 1 = 7x

are the 2 equations. (answer)

From the second equation

4y = 7x + 1

y = 7/4 x + 1/4

Substituting in the first equation:

7/4x + 1/4 + 16 = x^2

x^2 - 7/4 x - 16 - 1/4 = 0

x^2 - 7/4 x - 16 1/4 = 0

Multiplying though by 4

4x^2 - 7x - 65 = 0

Using the ac method to solve this 4 * -65 = -260 and we need factors of this to add up to -7. -20 and 13 look good so we write:

4x^2 - 20x + 13x - 65 = 0

Fatcor by grouping:

4x(x - 5) + 13(x - 5) = 0

(4x + 13)(x - 5) = 0

So the the roots are 5, -3.25

To find the values of y we substitute these values of x into the second equation:

x = 5: 4y - 1 = 7*542y = 36

y = 9.

x = -3.25:

4y - 1 = 7*-3.25

4y = (7 * -3.25) + 1

y = -5.44.

So the solutions are (5, 9) and (-3.25, -5.44) (Answer)