Graphing Systems Of Linear Equations Worksheet - Free Printable

Here are the solutions for each system of equations on the worksheet. I have worked through them step-by-step to find the correct answers.

Problem 1 (Top Left)

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

Step-by-step Solution:
Since both equations equal $y$, we can set them equal to each other:
$$2x - 4 = \frac{1}{3}x + 1$$

Subtract $\frac{1}{3}x$ from both sides:
$$\frac{6}{3}x - \frac{1}{3}x - 4 = 1$$
$$\frac{5}{3}x - 4 = 1$$

Add 4 to both sides:
$$\frac{5}{3}x = 5$$

Multiply by $\frac{3}{5}$ to isolate $x$:
$$x = 5 \cdot \frac{3}{5}$$
$$x = 3$$

Now, plug $x = 3$ back into the first equation to find $y$:
$$y = 2(3) - 4$$
$$y = 6 - 4$$
$$y = 2$$

Check with the second equation:
$$y = \frac{1}{3}(3) + 1 = 1 + 1 = 2$$ (It matches!)

Result: One Solution at (3, 2).

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Problem 2 (Top Right)

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

Step-by-step Solution:
Look closely at the second equation. The fraction $\frac{2}{4}$ simplifies to $\frac{1}{2}$.
So, the second equation is actually:
$$y = \frac{1}{2}x - 3$$

This is exactly the same as the first equation. Since the lines are identical, they lie on top of each other everywhere.

Result: Infinite Solutions.

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Problem 3 (Middle Left)

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

Step-by-step Solution:
Set the equations equal to each other:
$$-\frac{2}{3}x = x + 5$$

Subtract $x$ from both sides (think of $x$ as $\frac{3}{3}x$):
$$-\frac{2}{3}x - \frac{3}{3}x = 5$$
$$-\frac{5}{3}x = 5$$

Multiply both sides by $-\frac{3}{5}$:
$$x = 5 \cdot \left(-\frac{3}{5}\right)$$
$$x = -3$$

Plug $x = -3$ into the second equation to find $y$:
$$y = (-3) + 5$$
$$y = 2$$

Check with the first equation:
$$y = -\frac{2}{3}(-3) = 2$$ (It matches!)

Result: One Solution at (-3, 2).

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Problem 4 (Middle Right)

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

Step-by-step Solution:
Both equations are in slope-intercept form ($y = mx + b$).
* Equation 1 has a slope ($m$) of -1.
* Equation 2 has a slope ($m$) of -1.

Because the slopes are the same but the y-intercepts (3 and -1) are different, these lines are parallel. Parallel lines never cross.

Result: No Solution.

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Problem 5 (Bottom Left)

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

Step-by-step Solution:
Let's simplify the first equation to look like the second one. Divide every term in the first equation by 2:
$$\frac{2y}{2} = \frac{4x}{2} + \frac{4}{2}$$
$$y = 2x + 2$$

Now compare the two equations:
1) $y = 2x + 2$
2) $y = 2x + 1$

They both have a slope of 2, but different y-intercepts (2 and 1). This means they are parallel lines that will never touch.

Result: No Solution.

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Problem 6 (Bottom Right)

Equations:
1) $y = -2x - 3$
2) $4x + y = -1$

Step-by-step Solution:
First, rearrange the second equation into $y = \dots$ form so it's easier to work with. Subtract $4x$ from both sides:
$$y = -4x - 1$$

Now we have the system:
1) $y = -2x - 3$
2) $y = -4x - 1$

Set them equal to each other:
$$-2x - 3 = -4x - 1$$

Add $4x$ to both sides:
$$2x - 3 = -1$$

Add 3 to both sides:
$$2x = 2$$

Divide by 2:
$$x = 1$$

Plug $x = 1$ into the first equation to find $y$:
$$y = -2(1) - 3$$
$$y = -2 - 3$$
$$y = -5$$

Check with the second equation:
$$4(1) + (-5) = 4 - 5 = -1$$ (It matches!)

Result: One Solution at (1, -5).

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Final Answer:

Top Left:
One Solution: ( 3 , 2 )

Top Right:
Infinite Solutions

Middle Left:
One Solution: ( -3 , 2 )

Middle Right:
No Solution

Bottom Left:
No Solution

Bottom Right:
One Solution: ( 1 , -5 )