To solve the given system of equations, follow these steps:
1. Start with the two given equations:
[tex]\[
y = x^2 - 2x - 19
\][/tex]
[tex]\[
y + 4x = 5
\][/tex]
2. Rearrange the second equation to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 5 - 4x
\][/tex]
3. Substitute [tex]\( y = 5 - 4x \)[/tex] into the first equation:
[tex]\[
5 - 4x = x^2 - 2x - 19
\][/tex]
4. Bring all terms to one side of the equation to form a quadratic equation:
[tex]\[
x^2 - 2x - 19 + 4x - 5 = 0
\][/tex]
Simplify the equation:
[tex]\[
x^2 + 2x - 24 = 0
\][/tex]
5. Solve the quadratic equation [tex]\( x^2 + 2x - 24 = 0 \)[/tex]:
- To find the roots of the quadratic equation, use the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -24 \)[/tex].
6. Calculate the discriminant:
[tex]\[
b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-24) = 4 + 96 = 100
\][/tex]
7. Find the square root of the discriminant:
[tex]\[
\sqrt{100} = 10
\][/tex]
8. Compute the roots:
[tex]\[
x_1 = \frac{{-2 + 10}}{2} = \frac{8}{2} = 4
\][/tex]
[tex]\[
x_2 = \frac{{-2 - 10}}{2} = \frac{-12}{2} = -6
\][/tex]
9. Calculate the corresponding [tex]\( y \)[/tex] values for both [tex]\( x \)[/tex] values using [tex]\( y = 5 - 4x \)[/tex]:
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = 5 - 4 \cdot 4 = 5 - 16 = -11
\][/tex]
- For [tex]\( x = -6 \)[/tex]:
[tex]\[
y = 5 - 4 \cdot (-6) = 5 + 24 = 29
\][/tex]
10. Thus, the pair of points representing the solution set of this system of equations is [tex]\( (4, -11) \)[/tex] and [tex]\( (-6, 29) \)[/tex].
The correct answer for the solution set is:
[tex]\[
(4, -11)
\][/tex]
