To solve the given system of equations by using graphing, follow these steps:
### Step 1: Write Down the Equations
The system of equations is:
[tex]\[
\left\{\begin{array}{l}
y = \frac{1}{4}x + 3 \\
y = -\frac{1}{2}x
\end{array}\right.
\][/tex]
### Step 2: Determine the Slope and Intercept of Each Line
For [tex]\( y = \frac{1}{4}x + 3 \)[/tex]:
- Slope (m) = [tex]\( \frac{1}{4} \)[/tex]
- y-intercept (b) = 3
For [tex]\( y = -\frac{1}{2}x \)[/tex]:
- Slope (m) = [tex]\( -\frac{1}{2} \)[/tex]
- y-intercept (b) = 0
### Step 3: Draw the Graph of Each Line
1. Graph [tex]\( y = \frac{1}{4}x + 3 \)[/tex]:
- Start at the y-intercept (0, 3).
- Use the slope to find another point. With a slope of [tex]\( \frac{1}{4} \)[/tex], you rise 1 unit and run 4 units. From (0, 3), moving right 4 units, you move up 1 unit to the point (4, 4).
- Plot these points and draw a line through them.
2. Graph [tex]\( y = -\frac{1}{2}x \)[/tex]:
- Start at the y-intercept (0, 0).
- Use the slope to find another point. With a slope of [tex]\( -\frac{1}{2} \)[/tex], you rise -1 unit (or fall 1 unit) and run 2 units. From (0, 0), moving right 2 units, you move down 1 unit to the point (2, -1).
- Plot these points and draw a line through them.
### Step 4: Find the Intersection Point
The intersection point of the two lines is where they cross each other on the graph. By observing the graph, the lines intersect at the point (-4, 2).
### Step 5: Verify the Intersection Point
You can check this by substituting [tex]\( x = -4 \)[/tex] into both equations to verify that they give the same [tex]\( y \)[/tex]-value.
For [tex]\( y = \frac{1}{4}x + 3 \)[/tex]:
[tex]\[
y = \frac{1}{4}(-4) + 3 = -1 + 3 = 2
\][/tex]
For [tex]\( y = -\frac{1}{2}x \)[/tex]:
[tex]\[
y = -\frac{1}{2}(-4) = 2
\][/tex]
Both equations give [tex]\( y = 2 \)[/tex] when [tex]\( x = -4 \)[/tex].
### Final Answer
The solution to the system of equations is the point [tex]\((-4, 2)\)[/tex].
