Question

on the set of axes below, solve the following system of equations graphically and state the of all points in the solution set.
$y = -x^2 - 10x - 17$
$2x - 2y = -14$
you can move the parabola by dragging the dots. graph the line by clicking twice.

Explanation:

Step1: Rewrite the linear equation

First, we simplify the linear equation \(2x - 2y=-14\). Divide both sides by 2: \(x - y=-7\), then solve for \(y\): \(y=x + 7\).

Step2: Find intersection points (graphical method logic)

The solution to the system is the intersection points of the parabola \(y=-x^{2}-10x - 17\) and the line \(y=x + 7\). To find these points, we can also solve the system algebraically (as a check) by setting the two equations equal:
\(-x^{2}-10x - 17=x + 7\)
Rearrange to standard quadratic form:
\(-x^{2}-11x - 24 = 0\)
Multiply both sides by - 1: \(x^{2}+11x + 24=0\)
Factor the quadratic: \((x + 3)(x+8)=0\)
So \(x=-3\) or \(x = - 8\)

Step3: Find corresponding y - values

For \(x=-3\), substitute into \(y=x + 7\): \(y=-3 + 7=4\)
For \(x=-8\), substitute into \(y=x + 7\): \(y=-8 + 7=-1\)
(Graphically, we would plot the parabola \(y=-x^{2}-10x - 17\) (which has a vertex at \(x =-\frac{b}{2a}=-\frac{-10}{2\times(-1)}=-5\), \(y=-(-5)^{2}-10\times(-5)-17=-25 + 50-17 = 8\)) and the line \(y=x + 7\) (with slope 1 and y - intercept 7) and find their intersection points.)

Answer:

The solution set is \(\{(-8,-1),(-3,4)\}\)