Systems of Linear Equations Worksheets with Answer Key - Free Printable

Here are the step-by-step solutions for each problem on the worksheet using the substitution method.

Problem 1

$$ -2x + y = 5 $$
$$ 6x - 3y = -9 $$

Step 1: Solve the first equation for $y$.
Add $2x$ to both sides:
$$ y = 2x + 5 $$

Step 2: Substitute $(2x + 5)$ for $y$ in the second equation.
$$ 6x - 3(2x + 5) = -9 $$

Step 3: Solve for $x$.
Distribute the $-3$:
$$ 6x - 6x - 15 = -9 $$
Combine like terms ($6x - 6x = 0$):
$$ -15 = -9 $$

Conclusion: Since $-15$ does not equal $-9$, this is a false statement. This means there is no solution. The lines are parallel.

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Problem 2

$$ y = x + 4 $$
$$ 3x - 4y = -19 $$

Step 1: The first equation is already solved for $y$.
$$ y = x + 4 $$

Step 2: Substitute $(x + 4)$ for $y$ in the second equation.
$$ 3x - 4(x + 4) = -19 $$

Step 3: Solve for $x$.
Distribute the $-4$:
$$ 3x - 4x - 16 = -19 $$
Combine like terms:
$$ -x - 16 = -19 $$
Add $16$ to both sides:
$$ -x = -3 $$
Divide by $-1$:
$$ x = 3 $$

Step 4: Substitute $x = 3$ back into the first equation to find $y$.
$$ y = 3 + 4 $$
$$ y = 7 $$

Solution: $(3, 7)$

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Problem 3

$$ -4x + y = 6 $$
$$ -5x - y = 21 $$

Step 1: Solve the first equation for $y$.
Add $4x$ to both sides:
$$ y = 4x + 6 $$

Step 2: Substitute $(4x + 6)$ for $y$ in the second equation.
$$ -5x - (4x + 6) = 21 $$

Step 3: Solve for $x$.
Distribute the negative sign:
$$ -5x - 4x - 6 = 21 $$
Combine like terms:
$$ -9x - 6 = 21 $$
Add $6$ to both sides:
$$ -9x = 27 $$
Divide by $-9$:
$$ x = -3 $$

Step 4: Substitute $x = -3$ back into the equation from Step 1.
$$ y = 4(-3) + 6 $$
$$ y = -12 + 6 $$
$$ y = -6 $$

Solution: $(-3, -6)$

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Problem 4

$$ 6x + 4y = -8 $$
$$ -6x + y = 28 $$

Step 1: Solve the second equation for $y$ (it's the easiest one).
Add $6x$ to both sides:
$$ y = 6x + 28 $$

Step 2: Substitute $(6x + 28)$ for $y$ in the first equation.
$$ 6x + 4(6x + 28) = -8 $$

Step 3: Solve for $x$.
Distribute the $4$:
$$ 6x + 24x + 112 = -8 $$
Combine like terms:
$$ 30x + 112 = -8 $$
Subtract $112$ from both sides:
$$ 30x = -120 $$
Divide by $30$:
$$ x = -4 $$

Step 4: Substitute $x = -4$ back into the equation from Step 1.
$$ y = 6(-4) + 28 $$
$$ y = -24 + 28 $$
$$ y = 4 $$

Solution: $(-4, 4)$

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Problem 5

$$ x = -2y $$
$$ x - 9 = 9 $$

Step 1: Solve the second equation for $x$.
Add $9$ to both sides:
$$ x = 18 $$

Step 2: Substitute $18$ for $x$ in the first equation.
$$ 18 = -2y $$

Step 3: Solve for $y$.
Divide by $-2$:
$$ y = -9 $$

Solution: $(18, -9)$

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Problem 6

$$ x + y = 3 $$
$$ x - y = -3 $$

Step 1: Solve the first equation for $x$.
Subtract $y$ from both sides:
$$ x = 3 - y $$

Step 2: Substitute $(3 - y)$ for $x$ in the second equation.
$$ (3 - y) - y = -3 $$

Step 3: Solve for $y$.
Combine like terms:
$$ 3 - 2y = -3 $$
Subtract $3$ from both sides:
$$ -2y = -6 $$
Divide by $-2$:
$$ y = 3 $$

Step 4: Substitute $y = 3$ back into the equation from Step 1.
$$ x = 3 - 3 $$
$$ x = 0 $$

Solution: $(0, 3)$

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Problem 7

$$ x = y - 4 $$
$$ -2x + 3y = 6 $$

Step 1: The first equation is already solved for $x$.
$$ x = y - 4 $$

Step 2: Substitute $(y - 4)$ for $x$ in the second equation.
$$ -2(y - 4) + 3y = 6 $$

Step 3: Solve for $y$.
Distribute the $-2$:
$$ -2y + 8 + 3y = 6 $$
Combine like terms ($-2y + 3y = y$):
$$ y + 8 = 6 $$
Subtract $8$ from both sides:
$$ y = -2 $$

Step 4: Substitute $y = -2$ back into the first equation.
$$ x = -2 - 4 $$
$$ x = -6 $$

Solution: $(-6, -2)$

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Problem 8

$$ -5x + 20 = 5y $$
$$ -2x + y = 7 $$

Step 1: Solve the second equation for $y$.
Add $2x$ to both sides:
$$ y = 2x + 7 $$

Step 2: Substitute $(2x + 7)$ for $y$ in the first equation.
$$ -5x + 20 = 5(2x + 7) $$

Step 3: Solve for $x$.
Distribute the $5$:
$$ -5x + 20 = 10x + 35 $$
Subtract $10x$ from both sides:
$$ -15x + 20 = 35 $$
Subtract $20$ from both sides:
$$ -15x = 15 $$
Divide by $-15$:
$$ x = -1 $$

Step 4: Substitute $x = -1$ back into the equation from Step 1.
$$ y = 2(-1) + 7 $$
$$ y = -2 + 7 $$
$$ y = 5 $$

Solution: $(-1, 5)$

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Final Answer:
1. No Solution
2. (3, 7)
3. (-3, -6)
4. (-4, 4)
5. (18, -9)
6. (0, 3)
7. (-6, -2)
8. (-1, 5)