Lesson 24: Solving Systems of Equations Using the Elimination Method

New Concept

When the elimination method is used, the two equations are added. The sum of one of the variables is $0$.

What’s next

Next, you’ll solve systems by adding equations, sometimes multiplying them by a constant to enable elimination.

Property

If adding or subtracting the equations directly does not eliminate a variable, you must alter them first. When one coefficient is a simple multiple of the other, you only need to multiply ONE equation by a constant.

By multiplying every single term on both sides of that equation by the chosen constant, you create an equivalent equation with the exact opposite coefficient needed for elimination.

Examples

• Example 1 (Multiplying One Equation): Solve the system $3x + y = 5$ and $2x - 3y = 7$.

The $y$ coefficients are $1$ and $-3$. To make them opposites, multiply the ENTIRE top equation by $3$:
$3(3x + y) = 3(5) \rightarrow 9x + 3y = 15$
Now, add this new equation to the bottom equation:
$(9x + 2x) + (3y - 3y) = 15 + 7$
$11x = 22 \rightarrow x = 2$

• Example 2 (Back-Substitution): Now that $x = 2$, substitute it back into the original, simplest equation ($3x + y = 5$):

$3(2) + y = 5 \rightarrow 6 + y = 5 \rightarrow y = -1$.
The solution is $(2, -1)$.

Explanation

You can think of an equation like a recipe. If a recipe makes 1 batch of cookies, multiplying every single ingredient by 3 gives you 3 batches, but it is still the exact same recipe! In algebra, multiplying the left and right sides by the same number keeps the line exactly the same, but it changes the numbers to fit your needs. The most common mistake is multiplying the variables but forgetting to multiply the constant on the other side of the equal sign.

Property

A system with infinitely many solutions is a dependent system. The two lines coincide, meaning they are the same line. When you try to solve the system, you will get a true statement, such as $0 = 0$.

Solve $\begin{cases} 2x - y = 3 \\ -6x + 3y = -9 \end{cases}$. Multiply the first equation by 3: $3(2x - y) = 3(3)$ gives $6x - 3y = 9$.
Add the new equation to the second original equation: $(6x - 3y) + (-6x + 3y) = 9 + (-9)$, which simplifies to $0 = 0$.
Since $0=0$ is always true, the system has infinitely many solutions and is dependent.

Imagine you're solving a mystery with two clues. If you realize both clues are telling you the exact same thing, you don't have one answer—you have infinite possibilities that fit! That's a dependent system. The equations are just different ways of describing the same line, so every point on it is a solution.

Property

A linear system with no solutions is an inconsistent system. The graphs of the equations are parallel lines that never intersect. When solving, you will arrive at a false statement, such as $0 = 64$.

Solve $\begin{cases} 3x - 4y = -24 \\ 6x - 8y = 16 \end{cases}$. Multiply the first equation by -2: $-2(3x - 4y) = -2(-24)$ gives $-6x + 8y = 48$.
Add the new equation to the second original equation: $(-6x + 8y) + (6x - 8y) = 48 + 16$, which simplifies to $0 = 64$.
Since $0=64$ is a false statement, the system has no solution and is inconsistent.

This is like getting a math puzzle that leads to a contradiction. If you do everything right but end up with a nonsensical statement like $0 = 10$, the system is telling you something important. The two lines are parallel and will never, ever cross. There is no point in the universe that can make both equations true.