Question

What is the apparent solution to the system of equations?
3= 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 rewrite and solve both equations for easier graphing.

1. The first equation is:
\[
3 = \frac{1}{2}x + 2
\]

To isolate \( x \), subtract 2 from both sides:
\[
3 - 2 = \frac{1}{2}x
\]
\[
1 = \frac{1}{2}x
\]
Now, multiply both sides by 2:
\[
x = 2
\]

2. Next, substitute \( x = 2 \) into the second equation:
\[
y = 2x - 1
\]
\[
y = 2(2) - 1
\]
\[
y = 4 - 1 = 3
\]

Now we have the apparent solution:
\[
(x, y) = (2, 3)
\]

### Graphing the Equations

1. **Graph the first equation:**
- Rearranged from \( 3 = \frac{1}{2}x + 2 \) to \( y = \frac{1}{2}x + 2 - 3 \) or \( y = \frac{1}{2}x - 1 \).
- This line has a slope of \( \frac{1}{2} \) and a y-intercept of \(-1\).

2. **Graph the second equation:**
- From \( y = 2x - 1 \), the line has a slope of 2 and a y-intercept of \(-1\).

### To plot the graphs:
1. The first line (\( y = \frac{1}{2}x - 1 \)):
- A y-intercept at (0, -1).
- A point at \( (2, 2) \) (when \( x = 2 \)).

2. The second line (\( y = 2x - 1 \)):
- A y-intercept at (0, -1).
- A point at \( (2, 3) \).

### Plotting the solution:
Plot the solution point \( (2, 3) \) on the graph.

If you have access to graphing software or tools, you can use those to draw the lines corresponding to the equations and mark the point \( (2, 3) \) as the intersection of the two lines, confirming it is the apparent solution.