Solving Systems of Equations by Elimination

📝 Practice Worksheets

Graphing Linear Systems Graphing Systems Practice Systems of Inequalities

🛠️ Related Tool

Systems of Linear Equations Explorer

🔑 Key Concepts

Elimination combines two equations to remove one variable.
If a variable already has opposite coefficients, add the equations directly.
If the coefficients are not opposites, multiply one or both equations by a constant first.
Whatever operation is performed on an equation must be applied to every term in that equation.
After finding one variable, substitute its value into either original equation.
Write the solution as an ordered pair \((x,y)\).
The ordered pair represents the point where the two lines intersect.

✏️ Worked Examples — Solving Systems by Elimination

Example 1 — Opposite Coefficients

Solve the system:

\(\displaystyle \begin{cases} x+y=1\\ x-y=7 \end{cases} \)

Step 1: Add the equations.

The \(y\)-terms are opposites, so they eliminate immediately.

\(\begin{aligned} x+y&=1\\ x-y&=7\\ \hline 2x&=8 \end{aligned}\)

Step 2: Divide both sides by \(2\).

\(\displaystyle \frac{2x}{\color{red}{2}} = \frac{8}{\color{red}{2}} \)

\(x=4\)

Step 3: Substitute \(x=4\) into \(x+y=1\).

\(4+y=1\)

\(4+y\ {\color{red}{-4}} = 1\ {\color{red}{-4}}\)

\(y=-3\)

\(\boxed{(4,-3)}\)

Check:

\(4+(-3)=1\)

\(4-(-3)=7\)

Both equations are true, so the solution is \((4,-3)\).

Example 2 — Multiply Both Equations

Solve the system:

\(\displaystyle \begin{cases} 2x+3y=8\\ 3x-2y=-1 \end{cases} \)

Step 1: Create opposite coefficients for \(y\).

Multiply the first equation by \(2\) and the second equation by \(3\).

\({\color{red}{2}}(2x+3y=8)\)

\({\color{red}{3}}(3x-2y=-1)\)

\(\begin{aligned} 4x+6y&=16\\ 9x-6y&=-3 \end{aligned}\)

Step 2: Add the new equations.

\(\begin{aligned} 4x+6y&=16\\ 9x-6y&=-3\\ \hline 13x&=13 \end{aligned}\)

Step 3: Divide both sides by \(13\).

\(\displaystyle \frac{13x}{\color{red}{13}} = \frac{13}{\color{red}{13}} \)

\(x=1\)

Step 4: Substitute \(x=1\) into \(2x+3y=8\).

\(2(1)+3y=8\)

\(2+3y=8\)

\(2+3y\ {\color{red}{-2}} = 8\ {\color{red}{-2}}\)

\(3y=6\)

\(\displaystyle \frac{3y}{\color{red}{3}} = \frac{6}{\color{red}{3}} \)

\(y=2\)

\(\boxed{(1,2)}\)

The two lines intersect at \((1,2)\).

Example 3 — Multiply One Equation

Solve the system:

\(\displaystyle \begin{cases} 5x-2y=19\\ 3x+4y=1 \end{cases} \)

Step 1: Create opposite coefficients for \(y\).

Multiply the first equation by \(2\).

\({\color{red}{2}}(5x-2y=19)\)

\(10x-4y=38\)

The second equation remains unchanged.

\(3x+4y=1\)

Step 2: Add the equations.

\(\begin{aligned} 10x-4y&=38\\ 3x+4y&=1\\ \hline 13x&=39 \end{aligned}\)

Step 3: Divide both sides by \(13\).

\(\displaystyle \frac{13x}{\color{red}{13}} = \frac{39}{\color{red}{13}} \)

\(x=3\)

Step 4: Substitute \(x=3\) into \(5x-2y=19\).

\(5(3)-2y=19\)

\(15-2y=19\)

\(15-2y\ {\color{red}{-15}} = 19\ {\color{red}{-15}}\)

\(-2y=4\)

\(\displaystyle \frac{-2y}{\color{red}{-2}} = \frac{4}{\color{red}{-2}} \)

\(y=-2\)

\(\boxed{(3,-2)}\)

Check:

\(5(3)-2(-2)=19\)

\(3(3)+4(-2)=1\)

Both equations are true, so the solution is \((3,-2)\).

🧠 Math Vocabulary

System of equations: Two or more equations containing the same variables.
Elimination: A method that adds or subtracts equations to remove one variable.
Opposite coefficients: Coefficients with the same absolute value but different signs.
Equivalent equation: An equation with the same solutions as the original equation.
Solution of a system: The ordered pair that makes both equations true.
Intersection: The point where the graphs of the two equations meet.

💡 Main Idea

To solve a system using elimination, create opposite coefficients for one variable and add the equations so that variable disappears. Solve for the remaining variable, substitute its value into an original equation, and write the solution as the ordered pair where the two lines intersect.

📚 What You Should Already Know

Students should know how to solve multi-step equations, combine like terms, perform operations with positive and negative integers, use the distributive property, and multiply every term of an equation by a constant.

🚀 What Comes Next

Students will compare elimination with substitution and graphing, choose efficient methods for different systems, identify systems with no solution or infinitely many solutions, and apply systems to real-world problems.

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