Question

A system of linear equations includes the line that is created by the equation y = 0.5 x minus 1 and the line through the points (3, 1) and (–5, –7), shown below.

On a coordinate plane, points are at (negative 5, negative 7) and (3, 1).

What is the solution to the system of equations?
(–6, –4)
(0, –1)
(0, –2)
(2, 0)

Answer 1

To find the solution to the system of equations, we need to determine the point of intersection between the two lines represented by the equations.

1. The first line is given by the equation:
\[
y = 0.5x - 1
\]

2. Next, we need to determine the equation of the second line that passes through the points (3, 1) and (–5, –7).

We can use the two-point formula for the slope \(m\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-7 - 1}{-5 - 3} = \frac{-8}{-8} = 1
\]

Now that we have the slope, we can use point-slope form \(y - y_1 = m(x - x_1)\). Let's use point (3, 1):
\[
y - 1 = 1(x - 3)
\]
Simplifying this:
\[
y - 1 = x - 3 \implies y = x - 2
\]

So now we have the two equations:
1. \(y = 0.5x - 1\)
2. \(y = x - 2\)

Now we will set the right-hand sides of the equations equal to find the intersection:
\[
0.5x - 1 = x - 2
\]

To solve for \(x\):
\[
-1 + 2 = x - 0.5x \implies 1 = 0.5x
\]
\[
x = 2
\]

Now substitute \(x = 2\) back into one of the equations to find \(y\). We can use \(y = x - 2\):
\[
y = 2 - 2 = 0
\]

Thus, the point of intersection (solution to the system) is:
\[
(2, 0)
\]

Therefore, the solution to the system of equations is:
\[
\boxed{(2, 0)}
\]