Worksheet with systems of linear equations to solve using substitution.

Here are the step-by-step solutions for each system of linear equations using the substitution method.

System 1

$$2x - y = -13$$
$$x - 4y = -24$$

Step 1: Solve the second equation for $x$ because it has a coefficient of 1.
$$x = 4y - 24$$

Step 2: Substitute $(4y - 24)$ for $x$ in the first equation.
$$2(4y - 24) - y = -13$$

Step 3: Solve for $y$.
$$8y - 48 - y = -13$$
$$7y - 48 = -13$$
$$7y = 35$$
$$y = 5$$

Step 4: Substitute $y = 5$ back into the equation from Step 1 to find $x$.
$$x = 4(5) - 24$$
$$x = 20 - 24$$
$$x = -4$$

Solution: $x = -4, y = 5$

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System 2

$$y + 4x = -22$$
$$-2x + y = 26$$

Step 1: Solve the second equation for $y$.
$$y = 2x + 26$$

Step 2: Substitute $(2x + 26)$ for $y$ in the first equation.
$$(2x + 26) + 4x = -22$$

Step 3: Solve for $x$.
$$6x + 26 = -22$$
$$6x = -48$$
$$x = -8$$

Step 4: Substitute $x = -8$ back into the equation from Step 1 to find $y$.
$$y = 2(-8) + 26$$
$$y = -16 + 26$$
$$y = 10$$

Solution: $x = -8, y = 10$

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System 3

$$5x + y = -19$$
$$y - x = 11$$

Step 1: Solve the second equation for $y$.
$$y = x + 11$$

Step 2: Substitute $(x + 11)$ for $y$ in the first equation.
$$5x + (x + 11) = -19$$

Step 3: Solve for $x$.
$$6x + 11 = -19$$
$$6x = -30$$
$$x = -5$$

Step 4: Substitute $x = -5$ back into the equation from Step 1 to find $y$.
$$y = -5 + 11$$
$$y = 6$$

Solution: $x = -5, y = 6$

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System 4

$$2x + y = 20$$
$$6x - 5y = 12$$

Step 1: Solve the first equation for $y$.
$$y = -2x + 20$$

Step 2: Substitute $(-2x + 20)$ for $y$ in the second equation.
$$6x - 5(-2x + 20) = 12$$

Step 3: Solve for $x$.
$$6x + 10x - 100 = 12$$
$$16x - 100 = 12$$
$$16x = 112$$
$$x = 7$$

Step 4: Substitute $x = 7$ back into the equation from Step 1 to find $y$.
$$y = -2(7) + 20$$
$$y = -14 + 20$$
$$y = 6$$

Solution: $x = 7, y = 6$

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System 5

$$x = -3y - 14$$
$$5y + 4x = -21$$

Step 1: The first equation is already solved for $x$.
$$x = -3y - 14$$

Step 2: Substitute $(-3y - 14)$ for $x$ in the second equation.
$$5y + 4(-3y - 14) = -21$$

Step 3: Solve for $y$.
$$5y - 12y - 56 = -21$$
$$-7y - 56 = -21$$
$$-7y = 35$$
$$y = -5$$

Step 4: Substitute $y = -5$ back into the first equation to find $x$.
$$x = -3(-5) - 14$$
$$x = 15 - 14$$
$$x = 1$$

Solution: $x = 1, y = -5$

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System 6

$$x - 5y = -18$$
$$4y + 3x = 41$$

Step 1: Solve the first equation for $x$.
$$x = 5y - 18$$

Step 2: Substitute $(5y - 18)$ for $x$ in the second equation.
$$4y + 3(5y - 18) = 41$$

Step 3: Solve for $y$.
$$4y + 15y - 54 = 41$$
$$19y - 54 = 41$$
$$19y = 95$$
$$y = 5$$

Step 4: Substitute $y = 5$ back into the equation from Step 1 to find $x$.
$$x = 5(5) - 18$$
$$x = 25 - 18$$
$$x = 7$$

Solution: $x = 7, y = 5$

Final Answer:
1. $x = -4, y = 5$
2. $x = -8, y = 10$
3. $x = -5, y = 6$
4. $x = 7, y = 6$
5. $x = 1, y = -5$
6. $x = 7, y = 5$