Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
y = x² + 4x - 61
7 = x - y
answer
two solutions
and
submit answer

Explanation:

Step1: Express y from the second equation

From \( 7 = x - y \), we can rewrite it as \( y = x - 7 \).

Step2: Substitute y into the first equation

Substitute \( y = x - 7 \) into \( y = x^2 + 4x - 61 \), we get:
\( x - 7 = x^2 + 4x - 61 \)

Step3: Rearrange into standard quadratic form

Rearrange the equation: \( x^2 + 4x - 61 - x + 7 = 0 \)
Simplify to: \( x^2 + 3x - 54 = 0 \)

Step4: Factor the quadratic equation

Factor \( x^2 + 3x - 54 \): we need two numbers that multiply to -54 and add to 3. The numbers are 9 and -6.
So, \( (x + 9)(x - 6) = 0 \)

Step5: Solve for x

Set each factor equal to zero:
\( x + 9 = 0 \) gives \( x = -9 \)
\( x - 6 = 0 \) gives \( x = 6 \)

Step6: Find corresponding y values

For \( x = -9 \), substitute into \( y = x - 7 \): \( y = -9 - 7 = -16 \)
For \( x = 6 \), substitute into \( y = x - 7 \): \( y = 6 - 7 = -1 \)

Answer:

\((-9, -16)\) and \((6, -1)\)