Grade 8 / Solving Systems of Equations 2 / 6

Simultaneous Equations ② Elimination Method ① (Equal or Opposite Coefficients)

Simultaneous Equations ② Elimination Method ①

The elimination method solves equations by adding or subtracting the two equations to eliminate one variable. It is easiest when the coefficients of the variable to be eliminated are equal or have opposite signs.

Equal Coefficients (Eliminate by Subtracting)

Example ①
{ x+y=7 …①、x−y=1 …②
Answer:①−② → 2y=6 → y=3. Substitute into ①: x=4. (x,y)=(4,3)

Opposite-Sign Coefficients (Eliminate by Adding)

Example ②
{ 5x+y=11 …①、3x−y=5 …②
Answer:①+② → 8x=16 → x=2. Substitute into ①: y=1. (x,y)=(2,1)

Key Points

Equal coefficients → subtract to eliminate
Opposite-sign coefficients → add to eliminate
After elimination, the result is a linear equation in one variable
Substitute the value found into either equation to find the other variable

Practice Problems

Solve { x+y=9、x−y=3 using the elimination method.
Solve { 2x+3y=11、4x−3y=1 using the elimination method.
Solve { 3x+y=10、x−y=2 using the elimination method.

Answers and Explanations

Answer:(x,y)=(6,3)。①−②：2y=6→y=3。Substitute into ① to get x=6. Therefore (6,3)。
Answer:(x,y)=(2,7/3)。Add the equations: 6x=12→x=2. Substitute: 4+3y=11→y=7/3。
Answer:(x,y)=(3,1)。Add the equations: 4x=12→x=3. Substitute to get y=1。

Simultaneous Equations ① What Are Simultaneous Equations?

Simultaneous Equations ③ Elimination Method ② (Matching Coefficients)