Question

Graph and then find the solution for
y = -1/4x + 8 and y= 1/2 x + 2
.

Answer 1

To solve the system of equations given by \( y = -\frac{1}{4}x + 8 \) and \( y = \frac{1}{2}x + 2 \), we can first graph the two lines and then find the point of intersection.

1. **Graphing the equations:**

- For \( y = -\frac{1}{4}x + 8 \):
- The y-intercept is \( 8 \). This means the line crosses the y-axis at \( (0, 8) \).
- To find another point, you can choose \( x = 4 \):
\[
y = -\frac{1}{4}(4) + 8 = -1 + 8 = 7 \text{, thus the point (4, 7)}
\]
- So we have two points: \( (0, 8) \) and \( (4, 7) \).

- For \( y = \frac{1}{2}x + 2 \):
- The y-intercept is \( 2 \). This means this line crosses the y-axis at \( (0, 2) \).
- To find another point, you can choose \( x = 4 \):
\[
y = \frac{1}{2}(4) + 2 = 2 + 2 = 4 \text{, thus the point (4, 4)}
\]
- So we have two points: \( (0, 2) \) and \( (4, 4) \).

2. **Finding where the lines intersect:**

To find the solution algebraically, set the equations equal to each other:

\[
-\frac{1}{4}x + 8 = \frac{1}{2}x + 2
\]

Now, eliminate the fractions by multiplying the entire equation by \( 4 \):

\[
-x + 32 = 2x + 8
\]

Rearranging gives:

\[
32 - 8 = 2x + x
\]

\[
24 = 3x
\]

So:

\[
x = 8
\]

Now substitute \( x = 8 \) back into either original equation to find \( y \):

Using \( y = \frac{1}{2}x + 2 \):

\[
y = \frac{1}{2}(8) + 2 = 4 + 2 = 6
\]

Thus, the solution to the system of equations is:

\[
\boxed{(8, 6)}
\]

This indicates that the lines intersect at the point \( (8, 6) \).