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.