The Elimination Method - Solving A System Of Equations

📝 Practice Worksheets

Systems Using Substitution Graphing Systems of Equations Linear Systems Assessment

🛠️ Related Tool

Systems of Linear Equations Explorer

🔑 Key Concepts

Elimination combines two equations so that one variable disappears.
Add the equations when one variable already has opposite coefficients.
Multiply one or both equations when opposite coefficients must be created first.
Apply a multiplier to every term on both sides of an equation.
After finding one variable, substitute its value into either original equation.
Write the solution as an ordered pair \((x,y)\).
Check the ordered pair in both original equations.

✏️ Worked Examples — Solving Systems Using Elimination

Example 1 — Add the Equations

Solve the system:

\(\displaystyle \begin{cases} 4x+8y=20\\ -4x+2y=-30 \end{cases} \)

Step 1: Identify the opposite coefficients.

The \(x\)-coefficients are \(4\) and \(-4\), so the \(x\)-terms eliminate when the equations are added.

\(\begin{aligned} 4x+8y&=20\\ -4x+2y&=-30\\ \hline 10y&=-10 \end{aligned}\)

Step 2: Divide both sides by \(10\).

\(\displaystyle \frac{10y}{\color{red}{10}} = \frac{-10}{\color{red}{10}} \)

\(y=-1\)

Step 3: Substitute \(y=-1\) into \(4x+8y=20\).

\(4x+8(-1)=20\)

\(4x-8=20\)

\(4x-8\ {\color{red}{+8}} = 20\ {\color{red}{+8}}\)

\(4x=28\)

\(\displaystyle \frac{4x}{\color{red}{4}} = \frac{28}{\color{red}{4}} \)

\(x=7\)

\(\boxed{(7,-1)}\)

Check:

\(4(7)+8(-1)=20\)

\(-4(7)+2(-1)=-30\)

Both equations are true, so the solution is \((7,-1)\).

Example 2 — Multiply Both Equations

Solve the system:

\(\displaystyle \begin{cases} 3x-2y=2\\ 5x-5y=10 \end{cases} \)

Step 1: Create opposite \(x\)-coefficients.

Multiply the first equation by \(5\) and the second equation by \(-3\).

\({\color{red}{5}}(3x-2y=2)\)

\({\color{red}{-3}}(5x-5y=10)\)

\(\begin{aligned} 15x-10y&=10\\ -15x+15y&=-30 \end{aligned}\)

Step 2: Add the equations.

\(\begin{aligned} 15x-10y&=10\\ -15x+15y&=-30\\ \hline 5y&=-20 \end{aligned}\)

Step 3: Divide both sides by \(5\).

\(\displaystyle \frac{5y}{\color{red}{5}} = \frac{-20}{\color{red}{5}} \)

\(y=-4\)

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

\(3x-2(-4)=2\)

\(3x+8=2\)

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

\(3x=-6\)

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

\(x=-2\)

\(\boxed{(-2,-4)}\)

Check:

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

\(5(-2)-5(-4)=10\)

Both equations are true, so the solution is \((-2,-4)\).

Example 3 — Rewrite, Then Eliminate

Solve the system:

\(\displaystyle \begin{cases} 3+2x-y=0\\ -3-7y=10x \end{cases} \)

Step 1: Rewrite each equation in standard form.

Subtract \(3\) from both sides of the first equation.

\(3+2x-y\ {\color{red}{-3}} = 0\ {\color{red}{-3}}\)

\(2x-y=-3\)

Move \(10x\) to the left and add \(3\) to the right in the second equation.

\(-3-7y\ {\color{red}{-10x}} = 10x\ {\color{red}{-10x}}\)

\(-10x-7y=3\)

The equivalent system is:

\(\displaystyle \begin{cases} 2x-y=-3\\ -10x-7y=3 \end{cases} \)

Step 2: Multiply the first equation by \(5\).

\({\color{red}{5}}(2x-y=-3)\)

\(10x-5y=-15\)

Step 3: Add the equations.

\(\begin{aligned} 10x-5y&=-15\\ -10x-7y&=3\\ \hline -12y&=-12 \end{aligned}\)

Step 4: Divide both sides by \(-12\).

\(\displaystyle \frac{-12y}{\color{red}{-12}} = \frac{-12}{\color{red}{-12}} \)

\(y=1\)

Step 5: Substitute \(y=1\) into \(2x-y=-3\).

\(2x-1=-3\)

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

\(2x=-2\)

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

\(x=-1\)

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

Check:

\(3+2(-1)-1=0\)

\(-3-7(1)=10(-1)\)

Both equations are true, so the solution is \((-1,1)\).

🧠 Math Vocabulary

System of equations: Two or more equations containing the same variables.
Elimination: A method that combines equations to remove one variable.
Opposite coefficients: Coefficients with the same absolute value but opposite signs.
Equivalent system: A rewritten system that has the same solution as the original system.
Solution of a system: The ordered pair that makes both equations true.
Standard form: A linear equation written in the form \(Ax+By=C\).

💡 Main Idea

To solve a system using elimination, align the variables, create opposite coefficients when necessary, and combine the equations so one variable disappears. Solve for the remaining variable, substitute back, and write the solution as an ordered pair.

📚 What You Should Already Know

Students should know how to solve multi-step equations, combine like terms, perform integer operations, use the distributive property, rewrite equations in standard form, 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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