Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points. $y = 2x^2 + x - 75$ $3 = x - y$ answer two solutions and submit answer

Explanation:

Step1: Solve the second equation for y

From \( 3 = x - y \), we can rearrange it to get \( y = x - 3 \).

Step2: Substitute y into the first equation

Substitute \( y = x - 3 \) into \( y = 2x^2 + x - 75 \). So we have:
\( x - 3 = 2x^2 + x - 75 \)

Step3: Simplify the equation

Subtract \( x \) from both sides and add 3 to both sides:
\( 0 = 2x^2 - 72 \)
Divide both sides by 2:
\( x^2 - 36 = 0 \)

Step4: Factor the quadratic equation

Using the difference of squares formula \( a^2 - b^2=(a + b)(a - b) \), we get:
\( (x + 6)(x - 6)=0 \)

Step5: Solve for x

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

Step6: Find the corresponding y values

For \( x=-6 \), substitute into \( y=x - 3 \): \( y=-6 - 3=-9 \)
For \( x = 6 \), substitute into \( y=x - 3 \): \( y=6 - 3 = 3 \)

Answer:

\((-6,-9)\) and \((6,3)\)