Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction

\\(y = x^2 - 2x - 19\\)
\\(y + 4x = 5\\)

the pair of points representing the solution set of this system of equations is (-6, 29) and .

Explanation:

Express y in terms of x from the linear equation

We are given a system of equations:

1. \(y = x^2 - 2x - 19\)
2. \(y + 4x = 5\)

First, we isolate \(y\) in the second equation:

$$y = 5 - 4x$$

Substitute the expression for y into the quadratic equation

Since both expressions equal \(y\), we set them equal to each other:

$$x^2 - 2x - 19 = 5 - 4x$$

Solve the resulting quadratic equation for x

Rearrange the terms to form a standard quadratic equation \(ax^2 + bx + c = 0\):

$$x^2 - 2x + 4x - 19 - 5 = 0$$

$$x^2 + 2x - 24 = 0$$

We factor this quadratic equation by finding two numbers that multiply to \(-24\) and add to \(2\). These numbers are \(6\) and \(-4\):

$$(x + 6)(x - 4) = 0$$

This gives two possible solutions for \(x\):

$$x = -6 \quad \text{or} \quad x = 4$$

Find the corresponding y-coordinates

We substitute each \(x\)-value back into the linear equation \(y = 5 - 4x\) to find the corresponding \(y\)-coordinates.

For \(x = -6\):

$$y = 5 - 4(-6) = 5 + 24 = 29$$

This gives the first solution point: \((-6, 29)\).

For \(x = 4\):

$$y = 5 - 4(4) = 5 - 16 = -11$$

This gives the second solution point: \((4, -11)\).

Answer:

Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.

\(y = x^2 - 2x - 19\)
\(y + 4x = 5\)

The pair of points representing the solution set of this system of equations is (-6, 29) and <blank>(4, -11)</blank>.