Systems of Equations by Elimination - Free Printable

The task involves solving systems of linear equations using the Elimination Method. The Elimination Method is a technique where you manipulate the equations to eliminate one of the variables, allowing you to solve for the other variable. Once you have the value of one variable, you substitute it back into one of the original equations to find the value of the other variable.

Below, I will solve a few of the problems step by step to illustrate the process. If you need solutions for specific problems or all of them, let me know!

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Problem 1:

$$
\begin{aligned}
&2x + y = 7 \\
&x - y = 4
\end{aligned}
$$

#### Step 1: Write the equations.
$$
\begin{aligned}
1) & \quad 2x + y = 7 \\
2) & \quad x - y = 4
\end{aligned}
$$

#### Step 2: Add the two equations to eliminate \( y \).
$$
(2x + y) + (x - y) = 7 + 4
$$
$$
3x = 11
$$

#### Step 3: Solve for \( x \).
$$
x = \frac{11}{3}
$$

#### Step 4: Substitute \( x = \frac{11}{3} \) into one of the original equations to solve for \( y \). Use equation (2):
$$
x - y = 4
$$
$$
\frac{11}{3} - y = 4
$$
$$
-y = 4 - \frac{11}{3}
$$
$$
-y = \frac{12}{3} - \frac{11}{3}
$$
$$
-y = \frac{1}{3}
$$
$$
y = -\frac{1}{3}
$$

#### Step 5: Write the solution as an ordered pair.
$$
\boxed{\left( \frac{11}{3}, -\frac{1}{3} \right)}
$$

---

Problem 2:

$$
\begin{aligned}
&2x + 3y = 11 \\
&3x + 4y = 18
\end{aligned}
$$

#### Step 1: Write the equations.
$$
\begin{aligned}
1) & \quad 2x + 3y = 11 \\
2) & \quad 3x + 4y = 18
\end{aligned}
$$

#### Step 2: Eliminate one variable. To do this, make the coefficients of \( x \) or \( y \) the same in both equations. Let's eliminate \( x \). Multiply equation (1) by 3 and equation (2) by 2:
$$
\begin{aligned}
3 \cdot (2x + 3y) &= 3 \cdot 11 \\
6x + 9y &= 33 \quad \text{(Equation 3)}
\end{aligned}
$$
$$
\begin{aligned}
2 \cdot (3x + 4y) &= 2 \cdot 18 \\
6x + 8y &= 36 \quad \text{(Equation 4)}
\end{aligned}
$$

#### Step 3: Subtract equation (4) from equation (3) to eliminate \( x \):
$$
(6x + 9y) - (6x + 8y) = 33 - 36
$$
$$
6x + 9y - 6x - 8y = -3
$$
$$
y = -3
$$

#### Step 4: Substitute \( y = -3 \) into one of the original equations to solve for \( x \). Use equation (1):
$$
2x + 3y = 11
$$
$$
2x + 3(-3) = 11
$$
$$
2x - 9 = 11
$$
$$
2x = 20
$$
$$
x = 10
$$

#### Step 5: Write the solution as an ordered pair.
$$
\boxed{(10, -3)}
$$

---

Problem 3:

$$
\begin{aligned}
&-2x + 3y = 12 \\
&x - 2y = -7
\end{aligned}
$$

#### Step 1: Write the equations.
$$
\begin{aligned}
1) & \quad -2x + 3y = 12 \\
2) & \quad x - 2y = -7
\end{aligned}
$$

#### Step 2: Eliminate one variable. To do this, make the coefficients of \( x \) the same in both equations. Multiply equation (2) by 2:
$$
2 \cdot (x - 2y) = 2 \cdot (-7)
$$
$$
2x - 4y = -14 \quad \text{(Equation 3)}
$$

#### Step 3: Add equation (1) and equation (3) to eliminate \( x \):
$$
(-2x + 3y) + (2x - 4y) = 12 + (-14)
$$
$$
-2x + 2x + 3y - 4y = -2
$$
$$
-y = -2
$$
$$
y = 2
$$

#### Step 4: Substitute \( y = 2 \) into one of the original equations to solve for \( x \). Use equation (2):
$$
x - 2y = -7
$$
$$
x - 2(2) = -7
$$
$$
x - 4 = -7
$$
$$
x = -3
$$

#### Step 5: Write the solution as an ordered pair.
$$
\boxed{(-3, 2)}
$$

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General Approach Summary:

1. Write down the system of equations.
2. Choose a variable to eliminate by making its coefficients equal (usually by multiplying one or both equations by constants).
3. Add or subtract the equations to eliminate the chosen variable.
4. Solve for the remaining variable.
5. Substitute the value back into one of the original equations to solve for the other variable.
6. Write the solution as an ordered pair.

If you need solutions for more problems, please specify which ones!

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For now, the solutions for the first three problems are:
$$
\boxed{\left( \frac{11}{3}, -\frac{1}{3} \right), (10, -3), (-3, 2)}
$$