Question

the two lines graphed on the coordinate grid each represent an equation. which ordered pair represents a solution to both equations? (2, 9) (-9, 1) (6, -4) (-6, -7) (0, 5) (-3, -1) (-1, -3) (0, -2)

Explanation:

Step1: Recall the solution of a system of linear equations

The solution to a system of two linear equations (represented by two lines on a coordinate grid) is the point where the two lines intersect. So we need to find the intersection point of the two lines.

Step2: Analyze the lines' equations (or use the graph)

First, let's find the equations of the two lines.

For the first line (passing through \((-6, -7)\) and \((2, 9)\)):

The slope \(m_1=\frac{y_2 - y_1}{x_2 - x_1}=\frac{9 - (-7)}{2 - (-6)}=\frac{16}{8} = 2\).
Using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(2,9)\):
\(y - 9 = 2(x - 2)\)
\(y-9=2x - 4\)
\(y=2x + 5\)

For the second line (passing through \((-9, 1)\) and \((6, -4)\)):

The slope \(m_2=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-4 - 1}{6 - (-9)}=\frac{-5}{15}=-\frac{1}{3}\)
Using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-9,1)\):
\(y - 1=-\frac{1}{3}(x + 9)\)
\(y-1=-\frac{1}{3}x-3\)
\(y=-\frac{1}{3}x - 2\)

Step3: Find the intersection point

Set the two equations equal to each other:
\(2x + 5=-\frac{1}{3}x-2\)
Multiply through by 3 to eliminate the fraction:
\(6x+15=-x - 6\)
\(6x+x=-6 - 15\)
\(7x=-21\)
\(x=-3\)
Substitute \(x = - 3\) into \(y = 2x+5\):
\(y=2(-3)+5=-6 + 5=-1\)

So the intersection point (the solution to both equations) is \((-3,-1)\)

We can also check by looking at the graph. The two lines intersect at the point where \(x=-3\) and \(y = - 1\), which is the ordered pair \((-3,-1)\)

Answer:

\((-3, -1)\) (corresponding to the option \(\boldsymbol{(-3, -1)}\))