Systems of Equations Worksheets | Simultaneous Equations - Free Printable

It looks like you've uploaded a worksheet titled "Systems of Equations - Graphing", which asks students to:

1. Graph each system of equations on the provided coordinate planes.
2. Find the solution (the point of intersection) for each system.

Since I can't view images directly, I’ll help you solve this type of problem step-by-step based on what’s typical in such worksheets. Let's assume the problems are similar to common ones found in algebra curricula.

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🔹 General Approach to Solving Systems by Graphing

To solve a system of two linear equations by graphing:

1. Rewrite each equation in slope-intercept form:
$ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.

2. Graph each line:
- Plot the y-intercept.
- Use the slope to find another point.
- Draw the line through those points.

3. Find the point of intersection:
- The solution is the point where the two lines cross.
- This point satisfies both equations.

4. Write the solution as an ordered pair $ (x, y) $.

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Let’s go through each problem one at a time, assuming typical equations from such worksheets.

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✔ Problem 1:

Equations:
(1) $ y = x - 2 $
(2) $ y = -2x + 4 $

#### Step 1: Graph both lines
- For $ y = x - 2 $:
- Slope = 1, y-intercept = -2
- Plot (0, -2), then go up 1, right 1 → (1, -1), etc.

- For $ y = -2x + 4 $:
- Slope = -2, y-intercept = 4
- Plot (0, 4), then down 2, right 1 → (1, 2), etc.

#### Step 2: Find intersection
Set equations equal:
$$
x - 2 = -2x + 4 \\
x + 2x = 4 + 2 \\
3x = 6 \Rightarrow x = 2
$$
Plug back into first equation:
$$
y = 2 - 2 = 0
$$

✔ Solution: $ (2, 0) $

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✔ Problem 2:

Equations:
(1) $ y = 2x + 1 $
(2) $ y = -x - 2 $

#### Step 1: Graph
- Line 1: slope = 2, y-int = 1 → (0,1), (1,3)
- Line 2: slope = -1, y-int = -2 → (0,-2), (1,-3)

#### Step 2: Solve algebraically
$$
2x + 1 = -x - 2 \\
2x + x = -2 - 1 \\
3x = -3 \Rightarrow x = -1
$$
$$
y = 2(-1) + 1 = -2 + 1 = -1
$$

✔ Solution: $ (-1, -1) $

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✔ Problem 3:

Equations:
(1) $ y = \frac{1}{2}x + 3 $
(2) $ y = -\frac{1}{2}x + 5 $

#### Step 1: Graph
- Line 1: slope = 0.5, y-int = 3 → (0,3), (2,4)
- Line 2: slope = -0.5, y-int = 5 → (0,5), (2,4)

Wait — both pass through (2,4)? Let's check:

For $ x = 2 $:
- Eq1: $ y = 0.5(2) + 3 = 1 + 3 = 4 $
- Eq2: $ y = -0.5(2) + 5 = -1 + 5 = 4 $

So they intersect at $ (2, 4) $

✔ Solution: $ (2, 4) $

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✔ Problem 4:

Equations:
(1) $ y = -2x + 1 $
(2) $ y = x + 4 $

#### Step 1: Graph
- Line 1: slope = -2, y-int = 1 → (0,1), (1,-1)
- Line 2: slope = 1, y-int = 4 → (0,4), (1,5)

#### Step 2: Solve
$$
-2x + 1 = x + 4 \\
-2x - x = 4 - 1 \\
-3x = 3 \Rightarrow x = -1
$$
$$
y = -2(-1) + 1 = 2 + 1 = 3
$$

✔ Solution: $ (-1, 3) $

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✔ Problem 5:

Equations:
(1) $ y = \frac{1}{2}x + 2 $
(2) $ y = -x + 1 $

#### Step 1: Graph
- Line 1: slope = 0.5, y-int = 2 → (0,2), (2,3)
- Line 2: slope = -1, y-int = 1 → (0,1), (1,0)

#### Step 2: Solve
$$
\frac{1}{2}x + 2 = -x + 1 \\
\frac{1}{2}x + x = 1 - 2 \\
\frac{3}{2}x = -1 \\
x = -\frac{2}{3}
$$
Now plug into one equation:
$$
y = -(-\frac{2}{3}) + 1 = \frac{2}{3} + 1 = \frac{5}{3}
$$

✔ Solution: $ \left(-\frac{2}{3}, \frac{5}{3}\right) $

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✔ Problem 6:

Equations:
(1) $ y = -2x + 3 $
(2) $ y = 2x + 1 $

#### Step 1: Graph
- Line 1: slope = -2, y-int = 3 → (0,3), (1,1)
- Line 2: slope = 2, y-int = 1 → (0,1), (1,3)

#### Step 2: Solve
$$
-2x + 3 = 2x + 1 \\
-2x - 2x = 1 - 3 \\
-4x = -2 \Rightarrow x = \frac{1}{2}
$$
$$
y = 2(\frac{1}{2}) + 1 = 1 + 1 = 2
$$

✔ Solution: $ \left(\frac{1}{2}, 2\right) $

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## 📌 Final Answers Summary:

| Problem | Solution (x, y) |
|--------|------------------|
| 1 | $ (2, 0) $ |
| 2 | $ (-1, -1) $ |
| 3 | $ (2, 4) $ |
| 4 | $ (-1, 3) $ |
| 5 | $ \left(-\frac{2}{3}, \frac{5}{3}\right) $ |
| 6 | $ \left(\frac{1}{2}, 2\right) $ |

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✔ How to Fill in the Worksheet:

For each graph:
1. Carefully draw both lines using the slope and intercept.
2. Mark the point where they intersect.
3. Write the coordinates of that point in the blank space below the grid.

> Example: For Problem 1, mark the point (2, 0) and write: (2, 0)

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If your actual worksheet has different equations, please share them (type them out), and I’ll solve them exactly for you!

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