Solve Systems Of Linear Equations With Graphing - Worksheet - Free Printable

Let's solve each system of linear equations by graphing, and then determine the solution $(x, y)$ as the point where the two lines intersect. Since we can't actually drag points on this static image, I'll guide you through how to graph each pair of equations and find their intersection point algebraically (to verify), which corresponds to the solution.

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Problem 1:

$$
y = -\frac{4}{5}x + 5 \\
y = 1
$$

Step 1: Graph $y = 1$
This is a horizontal line passing through $y = 1$. It’s flat and crosses the y-axis at 1.

Step 2: Graph $y = -\frac{4}{5}x + 5$
- Slope: $-\frac{4}{5}$ (down 4, right 5)
- y-intercept: $(0, 5)$

Plot $(0, 5)$, then go down 4 and right 5 to get another point: $(5, 1)$. Draw the line.

Step 3: Find the intersection
The second line $y = 1$ intersects the first line when:
$$
1 = -\frac{4}{5}x + 5 \\
-\frac{4}{5}x = 1 - 5 = -4 \\
x = \frac{-4}{-\frac{4}{5}} = \frac{-4 \cdot 5}{-4} = 5
$$

So, the solution is:
$$
\boxed{(5, 1)}
$$

✔ Answer: $x = 5$, $y = 1$

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Problem 2:

$$
y = \frac{1}{4}x + 2 \\
y = x - 1
$$

Step 1: Graph $y = \frac{1}{4}x + 2$
- Slope: $\frac{1}{4}$ (up 1, right 4)
- y-intercept: $(0, 2)$

Plot $(0, 2)$, then go up 1, right 4 → $(4, 3)$

Step 2: Graph $y = x - 1$
- Slope: $1$
- y-intercept: $(0, -1)$

Plot $(0, -1)$, then go up 1, right 1 → $(1, 0)$

Step 3: Find intersection
Set equations equal:
$$
\frac{1}{4}x + 2 = x - 1 \\
2 + 1 = x - \frac{1}{4}x \\
3 = \frac{3}{4}x \\
x = \frac{3}{\frac{3}{4}} = 4
$$

Now plug back into one equation:
$$
y = 4 - 1 = 3
$$

So, the solution is:
$$
\boxed{(4, 3)}
$$

✔ Answer: $x = 4$, $y = 3$

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Problem 3:

$$
y = -\frac{8}{3}x + 3 \\
y = -\frac{4}{3}x - 1
$$

Step 1: Graph both lines

First line: $y = -\frac{8}{3}x + 3$
- Slope: $-\frac{8}{3}$ (down 8, right 3)
- y-intercept: $(0, 3)$

Second line: $y = -\frac{4}{3}x - 1$
- Slope: $-\frac{4}{3}$ (down 4, right 3)
- y-intercept: $(0, -1)$

Step 2: Solve algebraically
Set equal:
$$
-\frac{8}{3}x + 3 = -\frac{4}{3}x - 1 \\
3 + 1 = -\frac{4}{3}x + \frac{8}{3}x \\
4 = \frac{4}{3}x \\
x = 4 \cdot \frac{3}{4} = 3
$$

Now find $y$:
$$
y = -\frac{4}{3}(3) - 1 = -4 - 1 = -5
$$

So, the solution is:
$$
\boxed{(3, -5)}
$$

✔ Answer: $x = 3$, $y = -5$

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Problem 4:

$$
y = -3 \\
y = \frac{1}{4}x - 2
$$

Step 1: Graph $y = -3$
Horizontal line at $y = -3$

Step 2: Graph $y = \frac{1}{4}x - 2$
- Slope: $\frac{1}{4}$
- y-intercept: $(0, -2)$

Go up 1, right 4 → $(4, -1)$

Step 3: Find intersection
Set $y = -3$ in second equation:
$$
-3 = \frac{1}{4}x - 2 \\
-3 + 2 = \frac{1}{4}x \\
-1 = \frac{1}{4}x \\
x = -4
$$

So, the solution is:
$$
\boxed{(-4, -3)}
$$

✔ Answer: $x = -4$, $y = -3$

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✔ Final Answers Summary:

| Problem | Solution $(x, y)$ |
|--------|-------------------|
| 1 | $(5, 1)$ |
| 2 | $(4, 3)$ |
| 3 | $(3, -5)$ |
| 4 | $(-4, -3)$ |

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📌 How to Graph on the Worksheet:

For each problem:
1. Use the y-intercept to start.
2. Use the slope to plot a second point.
3. Draw the line.
4. The point where the two lines cross is the solution.

You can use these solutions to place the correct intersection point on the graph.

Let me know if you'd like a visual explanation or help plotting the graphs!