Question

solve the following system of equations.\

$$\begin{cases} y = x^2 + 7x - 3 \\\\ y = -4x - 21 \\end{cases}$$

if there is more than one solution, use the \or\ button.\
if applicable, click on
o solution\.\
\\((x, y) = (\square, \square)\\)

Explanation:

Step1: Set equations equal

Since $y=y$, equate the two expressions:
$x^2 + 7x - 3 = -4x - 21$

Step2: Rearrange to standard quadratic form

Bring all terms to left side:
$x^2 + 7x + 4x - 3 + 21 = 0$
$x^2 + 11x + 18 = 0$

Step3: Factor the quadratic

Find two factors of 18 that sum to 11:
$(x+2)(x+9) = 0$

Step4: Solve for x

Set each factor equal to 0:
$x+2=0 \implies x=-2$
$x+9=0 \implies x=-9$

Step5: Find y for each x

For $x=-2$: $y=-4(-2)-21=8-21=-13$
For $x=-9$: $y=-4(-9)-21=36-21=15$

Answer:

$(x,y)=(-2, -13)$ or $(x,y)=(-9, 15)$