Solving Systems of Equations by Elimination

Core Concept

• Definition: The elimination method solve systems by combining equations to remove one variable, making it easier to solve for the other.

Shortened Steps

1. Align: Line up variables and constants vertically (e.g., $x$ over $x$, $y$ over $y$).

2. Adjust: Multiply one or both equations by a constant to get identical or opposite coefficients for one variable.

3. Eliminate: Add or subtract the equations to solve for the first variable.

4. Substitute: Plug the result back into an original equation to find the second variable.

Representative Examples

1. $\begin{cases} 10x - 3y = 18 \ 2x + 3y = 6 \end{cases}$

   • Add equations to eliminate $3y$.

2. $\begin{cases} 4x + 3y = 13 \ x - y = -3 \end{cases}$

   • Multiply the second equation by $3$ to eliminate $y$.

Homework

Apply these four steps to the original list of systems to sharpen your skills.