Question

The system of equations y = 5/2 x - 4 and y = -1/2 x + 5 is shown on the graph below.

According to the graph, the solution to the system of equations is (_,_).

Answer 1

To find the solution to the system of equations, we need to determine the point where the two lines intersect. The equations given are:

1. \( y = \frac{5}{2}x - 4 \)
2. \( y = -\frac{1}{2}x + 5 \)

We can set these two equations equal to each other to find the x-coordinate of the intersection:

\[
\frac{5}{2}x - 4 = -\frac{1}{2}x + 5
\]

Now, we'll solve for \( x \):

1. Add \( \frac{1}{2}x \) to both sides:

\[
\frac{5}{2}x + \frac{1}{2}x - 4 = 5
\]

2. Combine the \( x \) terms on the left side:

\[
\frac{6}{2}x - 4 = 5
\]

This simplifies to:

\[
3x - 4 = 5
\]

3. Add \( 4 \) to both sides:

\[
3x = 9
\]

4. Divide by \( 3 \):

\[
x = 3
\]

Now that we have the x-coordinate, we can find the y-coordinate by substituting \( x = 3 \) back into either of the original equations. We'll use the first equation:

\[
y = \frac{5}{2}(3) - 4
\]

Calculating that:

\[
y = \frac{15}{2} - 4 = \frac{15}{2} - \frac{8}{2} = \frac{7}{2}
\]

So the intersection point (the solution to the system of equations) is:

\[
(3, \frac{7}{2})
\]

Thus, the solution to the system of equations is \( (3, \frac{7}{2}) \) or in numerical form \( (3, 3.5) \).

You can write the final answer as:

**The solution to the system of equations is (3, 3.5).**