Graphing Systems of Equations Worksheets - Math Monks - Free Printable

Let's solve each system of linear equations by graphing, as indicated in the activity. We’ll go through each problem step-by-step, rewrite equations in slope-intercept form if needed, find key points to plot, and determine the point of intersection — which is the solution to the system.

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Problem 1:

$$
\begin{align*}
(1)\quad & y = -x + 4 \\
(2)\quad & -3x + 2y = -2
\end{align*}
$$

#### Step 1: Rewrite second equation in slope-intercept form ($y = mx + b$)
$$
-3x + 2y = -2
$$
Add $3x$ to both sides:
$$
2y = 3x - 2
$$
Divide by 2:
$$
y = \frac{3}{2}x - 1
$$

So now we have:
- Line 1: $ y = -x + 4 $
- Line 2: $ y = \frac{3}{2}x - 1 $

#### Step 2: Graph both lines

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

- Line 2: $ y = \frac{3}{2}x - 1 $
- Slope: $1.5$, y-intercept: $-1$
- Points: (0, -1), (2, 2), (4, 5)

#### Step 3: Find point of intersection
From the points:
- Line 1 passes through (2, 2)
- Line 2 also passes through (2, 2)

✔ So the solution is $ (2, 2) $

> ✔ Answer for Problem 1: $ \boxed{(2, 2)} $

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Problem 2:

$$
\begin{align*}
(1)\quad & y = x - 1 \\
(2)\quad & x - y = 1
\end{align*}
$$

#### Step 1: Rewrite second equation
$$
x - y = 1
\Rightarrow -y = -x + 1
\Rightarrow y = x - 1
$$

Wait! Both equations are:
- $ y = x - 1 $
- $ y = x - 1 $

They are the same line!

#### Step 2: Interpretation
Since both equations represent the same line, there are infinitely many solutions — every point on the line is a solution.

✔ Answer for Problem 2: $ \boxed{\text{Infinitely many solutions}} $
*(The entire line $y = x - 1$)*

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Problem 3:

$$
\begin{align*}
(1)\quad & y = -5x - 3 \\
(2)\quad & 6x - 3y = -12
\end{align*}
$$

#### Step 1: Rewrite second equation in slope-intercept form
$$
6x - 3y = -12
$$
Subtract $6x$ from both sides:
$$
-3y = -6x - 12
$$
Divide by $-3$:
$$
y = 2x + 4
$$

Now we have:
- Line 1: $ y = -5x - 3 $
- Line 2: $ y = 2x + 4 $

#### Step 2: Graph both lines

- Line 1: $ y = -5x - 3 $
- Slope: $-5$, y-intercept: $-3$
- Points: (0, -3), (1, -8), (-1, 2)

- Line 2: $ y = 2x + 4 $
- Slope: $2$, y-intercept: $4$
- Points: (0, 4), (1, 6), (-1, 2)

Look at $(-1, 2)$:
- Plug into Line 1: $ y = -5(-1) - 3 = 5 - 3 = 2 $ → yes
- Plug into Line 2: $ y = 2(-1) + 4 = -2 + 4 = 2 $ → yes

✔ So they intersect at $ (-1, 2) $

> ✔ Answer for Problem 3: $ \boxed{(-1, 2)} $

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Problem 4:

$$
\begin{align*}
(1)\quad & 2x + 3y = -6 \\
(2)\quad & 2x + y = 2
\end{align*}
$$

#### Step 1: Rewrite both equations in slope-intercept form

Equation 1:
$$
2x + 3y = -6
\Rightarrow 3y = -2x - 6
\Rightarrow y = -\frac{2}{3}x - 2
$$

Equation 2:
$$
2x + y = 2
\Rightarrow y = -2x + 2
$$

Now we have:
- Line 1: $ y = -\frac{2}{3}x - 2 $
- Line 2: $ y = -2x + 2 $

#### Step 2: Graph both lines

- Line 1: $ y = -\frac{2}{3}x - 2 $
- Slope: $-\frac{2}{3}$, y-intercept: $-2$
- Points: (0, -2), (3, -4), (-3, 0)

- Line 2: $ y = -2x + 2 $
- Slope: $-2$, y-intercept: $2$
- Points: (0, 2), (1, 0), (2, -2)

#### Step 3: Find intersection

Try solving algebraically (since graphing may be imprecise):

Set equations equal:
$$
-\frac{2}{3}x - 2 = -2x + 2
$$

Multiply both sides by 3 to eliminate fraction:
$$
-2x - 6 = -6x + 6
$$

Add $6x$ to both sides:
$$
4x - 6 = 6
$$

Add 6:
$$
4x = 12 \Rightarrow x = 3
$$

Plug into one equation:
$$
y = -2(3) + 2 = -6 + 2 = -4
$$

Check in other:
$$
y = -\frac{2}{3}(3) - 2 = -2 - 2 = -4
$$

✔ Intersection at $ (3, -4) $

> ✔ Answer for Problem 4: $ \boxed{(3, -4)} $

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✔ Final Answers Summary:

1. $ \boxed{(2, 2)} $
2. $ \boxed{\text{Infinitely many solutions}} $
3. $ \boxed{(-1, 2)} $
4. $ \boxed{(3, -4)} $

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🎨 Coloring Activity Instructions (if applicable):

For each problem, you would:
- Graph both lines on the provided grid.
- Mark the point of intersection.
- Color the region or shape associated with that solution (as per coloring key, not shown here).

But since this is a text-based explanation, you can now plot these points and lines accurately on your graph paper.

Let me know if you'd like help plotting them visually!