Lesson 63: Solving Systems of Linear Equations by Elimination

New Concept

It is easier to eliminate one of the variables by combining the two equations using addition or subtraction.

What’s next

Next, you’ll apply this idea by adding, subtracting, and even multiplying entire equations to strategically remove variables and find the solution.

Property

When two linear equations have a variable with opposite coefficients (e.g., $ax$ and $-ax$), add the equations to eliminate that variable.

Explanation

Think of this as a mathematical disappearing act! When one variable in an equation is the exact opposite of its counterpart in the other equation, like a hero and its anti-hero ($+5x$ and $-5x$), you can simply add the two equations. The opposite terms cancel each other out, leaving a much simpler problem to solve.

Examples

• Given $3x + 4y = 10$ and $-3x + 2y = 8$, adding them yields $6y = 18$, so $y = 3$.
• For the system $x + y = 5$ and $-x + y = 1$, adding the equations results in $2y = 6$, so $y = 3$.
• In the system $8a - 5b = 11$ and $-8a - 3b = 5$, adding them gives $-8b = 16$, which simplifies to $b = -2$.

When two equations have perfectly opposite terms, adding them together can reveal the answer. This example uses the first key idea of this lesson, direct elimination.

Example Problem

Solve the system by elimination and check the answer: $4x + 3y = 5$ and $-4x + 5y = 11$.

Step-by-Step

1. The two equations have equal and opposite coefficients for the variable $x$, so we can add the equations to eliminate it.

2. Now, we solve for $y$.

3. Substitute $2$ for $y$ in the first original equation to solve for $x$.

4. The solution is $(-\frac{1}{4}, 2)$.
5. Check: Substitute $(-\frac{1}{4}, 2)$ into the second equation to verify.

By adding the equations, the x-terms cancelled out, making the problem much simpler. This additive elimination works whenever you spot opposite coefficients.

Property

When two linear equations have a variable with identical coefficients (e.g., $by$ and $by$), subtract one equation from the other to eliminate that variable.

Explanation

What happens when two terms are identical twins, like $3y$ and $3y$? You can eliminate one by subtracting the entire equation from the other! This is like having two identical songs playing and hitting mute on one. The duplicate term vanishes, simplifying the system so you can easily find the value of the remaining variable.

Examples

• Given $4x + 5y = 15$ and $2x + 5y = 9$, subtracting the second from the first gives $2x = 6$, so $x = 3$.
• For $8a - 2b = 12$ and $3a - 2b = 2$, subtracting the second equation results in $5a = 10$, so $a = 2$.
• In the system $10x + 3y = 20$ and $2x + 3y = 4$, subtraction yields $8x = 16$, which simplifies to $x = 2$.

Property

If no variables have the same or opposite coefficients, multiply one or both equations by a non-zero constant to create a pair of opposite coefficients.

Explanation

Sometimes, your equations aren't ready to cooperate and don't have matching or opposite terms. This is where you become the director! You can multiply one or even both equations by a strategic number to create the perfect set of opposites. Once you've engineered the cancellation, you just add the equations together and solve away.

Examples

• System: $3x + 2y = 7$ and $6x - 5y = 4$. Multiply the first equation by $-2$ to get $-6x - 4y = -14$ and eliminate $x$.
• System: $3a + 2b = 8$ and $2a - 5b = -1$. Multiply the first by $2$ and the second by $-3$ to get $6a+4b=16$ and $-6a+15b=3$.