Solve by Elimination
Solving a system of linear equations by elimination means adding or subtracting the two equations to cancel one variable, leaving a single equation you can solve directly. The method works by multiplying one or both equations by constants so a variable has equal-but-opposite coefficients, then adding the equations to eliminate it. Chapter 5 of OpenStax Elementary Algebra 2E presents elimination (also called the addition method) alongside substitution and graphing. Elimination is particularly powerful when neither variable is already isolated — it avoids the fractions that substitution can introduce.
Key Concepts
Property The Elimination Method is based on the Addition Property of Equality. When you add equal quantities to both sides of an equation, the results are equal. For any expressions $a, b, c$, and $d$, if $a = b$ and $c = d$, then $a+c = b+d$. To solve a system of equations by elimination, we start with both equations in standard form. We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.
Examples To solve the system $\begin{cases} x + y = 8 \\ x y = 4 \end{cases}$, we add the equations. The $y$ terms are opposites and eliminate, giving $2x=12$, so $x=6$. Substituting back, $6+y=8$, so $y=2$. The solution is $(6, 2)$. To solve $\begin{cases} 2x + y = 7 \\ 3x 2y = 0 \end{cases}$, multiply the first equation by 2 to make the $y$ coefficients opposites: $4x + 2y = 14$. Adding this to $3x 2y = 0$ gives $7x=14$, so $x=2$. Then $2(2)+y=7$, so $y=3$. The solution is $(2, 3)$. To solve $\begin{cases} 3x + 2y = 8 \\ 2x + 5y = 9 \end{cases}$, multiply the first equation by 2 and the second by $ 3$ to get $6x$ and $ 6x$. This gives $\begin{cases} 6x + 4y = 16 \\ 6x 15y = 27 \end{cases}$. Adding them yields $ 11y = 11$, so $y=1$. Then $3x+2(1)=8$, so $x=2$. The solution is $(2, 1)$.
Explanation This method adds two equations together. The goal is to make the coefficients of one variable opposites (like $5x$ and $ 5x$). When you add the equations, that variable cancels out, leaving a simple, one variable equation to solve.
Common Questions
How do you solve a system of equations by elimination?
Multiply one or both equations so that one variable has coefficients that are equal and opposite. Add the equations together to cancel that variable, solve for the remaining variable, then substitute back to find the other.
When should I use elimination instead of substitution?
Use elimination when neither equation has a variable with a coefficient of 1, because substitution would introduce fractions. Elimination is also faster when coefficients are already equal or easily made equal.
What do I do if neither variable cancels automatically?
Multiply one or both equations by constants to make the coefficients of one variable equal and opposite. For example, if one equation has 2x and the other has 3x, multiply the first by 3 and the second by 2 to get 6x in both, then subtract.
What happens if both variables cancel when I add the equations?
If both variables cancel and you get a true statement (like 0 = 0), the system has infinitely many solutions (dependent). If you get a false statement (like 0 = 5), there is no solution (inconsistent).
When do students learn the elimination method?
Elimination is an algebra 1 topic, covered in OpenStax Elementary Algebra 2E Chapter 5: Systems of Linear Equations.
What is the most common mistake when using elimination?
Forgetting to multiply every term in the equation by the constant, or making a sign error when adding negative terms.
How do I check the solution after using elimination?
Substitute both values (x, y) into both original equations and verify that both are true.