Question

Solve the system of linear equations by graphing.

y equals negative one half times x plus 3

y equals one half times x minus 2
one half comma negative 5
one half comma 5
5 comma one half
negative 5 comma one half

Answer 1

To solve the system of linear equations by graphing, we have the following two equations:

1. \( y = -\frac{1}{2}x + 3 \)
2. \( y = \frac{1}{2}x - 2 \)

### Step 1: Graphing the first equation
For the first equation \( y = -\frac{1}{2}x + 3 \):

- **Y-intercept**: When \( x = 0 \), \( y = 3 \). This gives the point (0, 3).
- **X-intercept**: When \( y = 0 \), \( 0 = -\frac{1}{2}x + 3 \) implies \( \frac{1}{2}x = 3 \) or \( x = 6 \). This gives the point (6, 0).

Now plot these points and draw a line through them.

### Step 2: Graphing the second equation
For the second equation \( y = \frac{1}{2}x - 2 \):

- **Y-intercept**: When \( x = 0 \), \( y = -2 \). This gives the point (0, -2).
- **X-intercept**: When \( y = 0 \), \( 0 = \frac{1}{2}x - 2 \) implies \( \frac{1}{2}x = 2 \) or \( x = 4 \). This gives the point (4, 0).

Plot these points and draw a line through them.

### Step 3: Finding the intersection point
After graphing both lines, find the point where they intersect, which is the solution to the system of equations.

### Step 4: Determine the coordinates
Now, we can check if the intersection point falls among the provided points:

1. \( \left(\frac{1}{2}, -5\right) \)
2. \( \left(\frac{1}{2}, 5\right) \)
3. \( (5, \frac{1}{2}) \)
4. \( (-5, \frac{1}{2}) \)

### Checking the coordinates:
1. For \( \left(\frac{1}{2}, -5\right) \):
- Substituting in \( y = -\frac{1}{2}x + 3 \):
\( -5 \stackrel{?}{=} -\frac{1}{2} \cdot \frac{1}{2} + 3 = -\frac{1}{4} + 3 = \frac{11}{4} \) (Not a solution)

2. For \( \left(\frac{1}{2}, 5\right) \):
- From both equations, test against:
- \( y = -\frac{1}{2}x + 3 \):
\( 5 \stackrel{?}{=} -\frac{1}{2} \cdot \frac{1}{2} + 3 \) (Not a solution)

3. For \( (5, \frac{1}{2}) \):
- From both equations:
- Test \( y = -\frac{1}{2}x + 3 \):
\( \frac{1}{2} = -\frac{1}{2}(5) + 3 \)
\( \frac{1}{2} = -\frac{5}{2} + 3 = \frac{1}{2} \) (This works!)

4. For \( (-5, \frac{1}{2}) \):
- Substituting into both equations shows that this point does not satisfy the equations.

### Conclusion:
The point \( (5, \frac{1}{2}) \) is a solution to the system of equations. Therefore, the solution to the system of linear equations by graphing is:

**(5, 1/2)**.