The substitution method solves a system by rearranging one equation into the form x= or y= and substituting it into the other equation. It is especially useful when one equation is already close to the form x= or y=.
Steps in the Substitution Method
① Rearrange one equation into the form x= or y=
② Substitute that equation into the other equation to create a linear equation in one variable
③ Solve it, then substitute the value found into the equation in ① to find the other variable
Example ①
{ y=2x−1 …①、3x+y=9 …②
Answer:Substitute ① into ②: 3x+(2x−1)=9 → 5x=10 → x=2. Substitute into ①: y=3. (2,3)
{ y=2x−1 …①、3x+y=9 …②
Answer:Substitute ① into ②: 3x+(2x−1)=9 → 5x=10 → x=2. Substitute into ①: y=3. (2,3)
Example ②
{ 2x+y=5 …①、x−y=1 …②
Answer:From ②, x=y+1. Substitute into ①: 2(y+1)+y=5 → 3y=3 → y=1. x=2. (2,1)
{ 2x+y=5 …①、x−y=1 …②
Answer:From ②, x=y+1. Substitute into ①: 2(y+1)+y=5 → 3y=3 → y=1. x=2. (2,1)
Key Points
- When the form x= or y= is visible, the substitution method is quick
- Do not forget parentheses when substituting
- The elimination and substitution methods give the same answer
Practice Problems
- Solve { y=3x、2x+y=10 using the substitution method.
- Solve { x=2y+3、x+y=9 using the substitution method.
- Solve { 3x−y=7、y=x+1 using the substitution method.
Answers and Explanations
- Answer:(x,y)=(2,6)。2x+3x=10→x=2,y=6。
- Answer:(x,y)=(7,2)。(2y+3)+y=9→3y=6→y=2,x=7。
- Answer:(x,y)=(4,5)。3x−(x+1)=7→2x=8→x=4,y=5。