Worksheet for solving systems of linear equations using the substitution method, with eight pairs of equations to solve.

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

1.
Equations:
$-2x + y = 5$
$6x - 3y = -9$

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 ($6x - 6x = 0$):
$-15 = -9$

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

2.
Equations:
$y = x + 4$
$3x - 4y = -19$

Step 1: The first equation is already solved for $y$. Substitute $(x + 4)$ for $y$ in the second equation.
$3x - 4(x + 4) = -19$

Step 2: Solve for $x$.
Distribute the $-4$:
$3x - 4x - 16 = -19$
Combine like terms:
$-x - 16 = -19$
Add $16$ to both sides:
$-x = -3$
Multiply by $-1$:
$x = 3$

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

Answer: $(3, 7)$

3.
Equations:
$-4x + y = 6$
$-5x - y = 21$

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.
$y = 4(-3) + 6$
$y = -12 + 6$
$y = -6$

Answer: $(-3, -6)$

4.
Equations:
$6x + 4y = -8$
$-6x + y = 28$

Step 1: Solve the second equation for $y$ (it looks easiest).
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.
$y = 6(-4) + 28$
$y = -24 + 28$
$y = 4$

Answer: $(-4, 4)$

5.
Equations:
$x = -2y$
$x - 9 = 9$

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 $-2$:
$y = -9$

Answer: $(18, -9)$

6.
Equations:
$x + y = 3$
$x - y = -3$

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.
$x = 3 - 3$
$x = 0$

Answer: $(0, 3)$

7.
Equations:
$x = y - 4$
$-2x + 3y = 6$

Step 1: The first equation is already solved for $x$. Substitute $(y - 4)$ for $x$ in the second equation.
$-2(y - 4) + 3y = 6$

Step 2: 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 3: Substitute $y = -2$ back into the first equation.
$x = -2 - 4$
$x = -6$

Answer: $(-6, -2)$

8.
Equations:
$-5x + 20 = 5y$
$-2x + y = 7$

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 $5$ on the right side:
$-5x + 20 = 10x + 35$
Add $5x$ to both sides:
$20 = 15x + 35$
Subtract $35$ from both sides:
$-15 = 15x$
Divide by $15$:
$x = -1$

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

Answer: $(-1, 5)$

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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. $(-1, 5)$