Graphing Systems of Equations Worksheets - Math Monks - Free Printable

Let's solve each system of equations by graphing, as requested. Since we can't draw on the image directly, I’ll guide you through how to graph each system and find the solution (the point where the lines intersect). Then, I'll provide the coordinates of the intersection point for each system.

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🔷 Problem 1:

Equations:
- $ y = 3x - 3 $
- $ y = -3 $

#### Step-by-step:
- The first equation is a line with slope 3 and y-intercept -3.
- The second equation is a horizontal line at $ y = -3 $.

To graph:
1. For $ y = 3x - 3 $:
- When $ x = 0 $, $ y = -3 $ → point: $ (0, -3) $
- When $ x = 1 $, $ y = 0 $ → point: $ (1, 0) $
- Plot these and draw the line.

2. For $ y = -3 $:
- Draw a horizontal line passing through $ y = -3 $.

Intersection:
Set $ 3x - 3 = -3 $
→ $ 3x = 0 $ → $ x = 0 $

So, the solution is $ (0, -3) $

✔ Answer: (0, -3)

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🔷 Problem 2:

Equations:
- $ -y - 3 + 4x = 0 $
- $ -4 = -3x - y $

#### Step 1: Rewrite both in slope-intercept form ($ y = mx + b $)

First equation:
$ -y - 3 + 4x = 0 $
→ $ -y = -4x + 3 $
→ $ y = 4x - 3 $

Second equation:
$ -4 = -3x - y $
→ $ y = -3x + 4 $ (add $ 3x $ to both sides, then add 4)

Now we have:
- $ y = 4x - 3 $
- $ y = -3x + 4 $

#### Graphing:
1. $ y = 4x - 3 $: slope = 4, y-intercept = -3
- Points: $ (0, -3), (1, 1) $

2. $ y = -3x + 4 $: slope = -3, y-intercept = 4
- Points: $ (0, 4), (1, 1) $

Wait! Both pass through $ (1, 1) $? Let's check:

- First: $ y = 4(1) - 3 = 1 $
- Second: $ y = -3(1) + 4 = 1 $

Yes! So they intersect at $ (1, 1) $

✔ Answer: (1, 1)

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🔷 Problem 3:

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

#### Step 1: Solve each for $ y $

First:
$ y = -2x + 1 $

Second:
$ -y = -2x + 3 $ → $ y = 2x - 3 $

Now:
- $ y = -2x + 1 $
- $ y = 2x - 3 $

#### Graphing:
1. $ y = -2x + 1 $: slope = -2, y-int = 1 → points: $ (0,1), (1,-1) $
2. $ y = 2x - 3 $: slope = 2, y-int = -3 → points: $ (0,-3), (1,-1) $

Wait — both go through $ (1, -1) $? Check:

- First: $ y = -2(1) + 1 = -1 $
- Second: $ y = 2(1) - 3 = -1 $

Yes! So they intersect at $ (1, -1) $

✔ Answer: (1, -1)

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🔷 Problem 4:

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

These are already in slope-intercept form.

#### Graphing:
1. $ y = -\frac{1}{4}x + 3 $:
- Slope = -1/4, y-int = 3
- Points: $ (0, 3), (4, 2) $

2. $ y = -\frac{3}{2}x - 2 $:
- Slope = -3/2, y-int = -2
- Points: $ (0, -2), (2, -5) $

Now find intersection by solving:
$$
-\frac{1}{4}x + 3 = -\frac{3}{2}x - 2
$$

Multiply both sides by 4 to eliminate denominators:
$$
-1x + 12 = -6x - 8
$$

Add $ 6x $ to both sides:
$$
5x + 12 = -8
$$

Subtract 12:
$$
5x = -20 \Rightarrow x = -4
$$

Now plug into one equation:
$ y = -\frac{1}{4}(-4) + 3 = 1 + 3 = 4 $

So, intersection at $ (-4, 4) $

✔ Answer: (-4, 4)

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✔ Final Answers:

1. $ (0, -3) $
2. $ (1, 1) $
3. $ (1, -1) $
4. $ (-4, 4) $

You can now plot each pair of lines on the grids provided and mark the intersection point accordingly.

Let me know if you'd like me to generate the graphs or explain how to sketch them step by step!