Question

solve the following system of equations.
$x^2 = y + 5$
$x + y = 7$
simplify your answer as much as possible.
if there is more than one solution, use the \or\ button.
$(x,y)=(square,square)$

Explanation:

Step1: Isolate y from linear equation

From $x + y = 7$, rearrange to get $y = 7 - x$.

Step2: Substitute y into quadratic equation

Substitute $y = 7 - x$ into $x^2 = y + 5$:
$x^2 = (7 - x) + 5$
Simplify the right-hand side:
$x^2 = 12 - x$
Rearrange into standard quadratic form:
$x^2 + x - 12 = 0$

Step3: Factor the quadratic equation

Factor $x^2 + x - 12$:
$(x + 4)(x - 3) = 0$
Solve for x:
$x + 4 = 0 \implies x = -4$
$x - 3 = 0 \implies x = 3$

Step4: Find corresponding y values

For $x = -4$: $y = 7 - (-4) = 11$
For $x = 3$: $y = 7 - 3 = 4$

Answer:

$(x,y)=(-4,11)$ or $(x,y)=(3,4)$