Question

On a coordinate plane, a line goes through (0, negative 5) and (5, 0).
A system of equations consists of a line s of the equation y = x - 5 that is graphed in orange, and a line t that passes through the points (0, 2) and (8, -4). The equation of line t is y = −3
4
x + 2. What is the solution to this system of linear equations?

(
,
)

Answer 1

To find the solution to the system of equations represented by the lines \( s: y = x - 5 \) and \( t: y = -\frac{3}{4}x + 2 \), we need to find the point where these two lines intersect. This is done by setting the two equations equal to each other:

\[
x - 5 = -\frac{3}{4}x + 2
\]

Now, let's solve for \( x \):

1. First, let's get all the \( x \) terms on one side. Add \(\frac{3}{4}x\) to both sides:
\[
x + \frac{3}{4}x - 5 = 2
\]

2. Combine the \( x \) terms. The left side can be rewritten as:
\[
\frac{4}{4}x + \frac{3}{4}x - 5 = 2
\]
\[
\frac{7}{4}x - 5 = 2
\]

3. Now, add 5 to both sides:
\[
\frac{7}{4}x = 7
\]

4. Multiply both sides by \(\frac{4}{7}\) to isolate \( x \):
\[
x = 7 \cdot \frac{4}{7} = 4
\]

Now that we have \( x = 4 \), we can substitute this back into either equation to find \( y \). We'll use the first equation \( y = x - 5 \):

\[
y = 4 - 5 = -1
\]

Thus, the solution to the system of equations is:

\[
(4, -1)
\]

So the answer is:
\[
\boxed{(4, -1)}
\]