Question

What is the apparent solution to the system of equations?
y = 1/2 x +2
y = 2x - 1
Graph the system of equations using the Line tool. Plot a point at the apparent solution to the system using the Point tool.

Answer 1

To find the apparent solution to the system of equations, we need to solve the equations simultaneously. The equations provided are:

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

To find the point where these lines intersect, we can set the right-hand sides of the equations equal to each other:

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

Now, we can solve for \( x \). First, let's eliminate the fraction by multiplying the entire equation by 2:

\[
x + 4 = 4x - 2
\]

Now, move \( 4x \) to the left side and add 2 to the left side:

\[
x + 4 + 2 = 4x
\]

Thus, we have:

\[
6 = 3x
\]

Dividing both sides by 3 gives:

\[
x = 2
\]

Now that we have \( x \), we can substitute it back into either equation to find \( y \). Let's substitute it into Equation 1:

\[
y = \frac{1}{2}(2) + 2 = 1 + 2 = 3
\]

Thus, the apparent solution to the system of equations is \( (2, 3) \).

### Graphing the Equations

To graph these lines:

1. For Equation 1 (\( y = \frac{1}{2}x + 2 \)):
- When \( x = 0 \), \( y = 2 \) (point: (0,2))
- When \( x = 2 \), \( y = 3 \) (point: (2,3))
- When \( x = 4 \), \( y = 4 \) (point: (4,4))

2. For Equation 2 (\( y = 2x - 1 \)):
- When \( x = 0 \), \( y = -1 \) (point: (0,-1))
- When \( x = 1 \), \( y = 1 \) (point: (1,1))
- When \( x = 2 \), \( y = 3 \) (point: (2,3))
- When \( x = 3 \), \( y = 5 \) (point: (3,5))

### Plotting Points

The lines will intersect at the point \( (2, 3) \), which can be plotted on the graph. The two lines will cross at this point.

### Using the Graphing Tool

If you're using a graphing tool:
- Use the Line tool to draw both lines based on the equations.
- Then use the Point tool to mark the intersection point, which you found to be \( (2, 3) \).

This method allows you to visualize the solution to the system graphically.