Question

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

Answer 1

To find the solution for the two equations \(y = \frac{1}{2}x + 2\) and \(y = -\frac{1}{4}x + 8\), we can follow these steps:

1. **Graph the equations**:
- For the first equation, \(y = \frac{1}{2}x + 2\):
- The y-intercept is 2 (which is the point (0, 2)).
- The slope is \(\frac{1}{2}\), which means for every 2 units you move to the right (increase x by 2), you move up 1 unit (increase y by 1).
- Another point can be found at \(x = 2\):
\[y = \frac{1}{2}(2) + 2 = 1 + 2 = 3\]
So another point is (2, 3).

- For the second equation, \(y = -\frac{1}{4}x + 8\):
- The y-intercept is 8 (which is the point (0, 8)).
- The slope is \(-\frac{1}{4}\), meaning for every 4 units you move to the right (increase x by 4), you move down 1 unit (decrease y by 1).
- Another point can be found at \(x = 4\):
\[y = -\frac{1}{4}(4) + 8 = -1 + 8 = 7\]
So another point is (4, 7).

2. **Plot the points and draw the lines**:
- Plot the points (0, 2) and (2, 3) for the first line and draw a line through them.
- Plot the points (0, 8) and (4, 7) for the second line and draw a line through them.

3. **Find the intersection point**:
- To find the solution algebrically, set the two equations equal to each other:
\[
\frac{1}{2}x + 2 = -\frac{1}{4}x + 8
\]

- Multiply every term by 4 to eliminate the fractions:
\[
4 \left(\frac{1}{2}x\right) + 4(2) = 4\left(-\frac{1}{4}x\right) + 4(8)
\]
\[
2x + 8 = -x + 32
\]

- Add \(x\) to both sides:
\[
2x + x + 8 = 32
\]
\[
3x + 8 = 32
\]

- Subtract 8 from both sides:
\[
3x = 24
\]

- Divide by 3:
\[
x = 8
\]

4. **Substitute \(x\) back into one of the original equations to find \(y\)**:
- Using the first equation:
\[
y = \frac{1}{2}(8) + 2 = 4 + 2 = 6
\]

So, the solution to the system of equations is \((8, 6)\).

### Summary:
- The intersection point is \((8, 6)\).
- The lines representing the two equations intersect at this point.