Simultaneous Equations ② Substitution Method
Learn the “substitution method,” in which one equation is substituted into the other. It is especially effective when one of the equations is in the form y = ~.
Steps for the Substitution Method
- Transform one equation into “y = (an expression in x)” or “x = (an expression in y)”
- Substitute that equation into the other equation
- Solve the equation with one variable
- Substitute the value found to determine the remaining variable
Example 1
{y＝2x－1 … ①
{3x＋y＝9 … ②
Solution:
Substitute ① into ②: 3x＋(2x－1)＝9
5x－1＝9
5x＝10 → x＝2
Substitute into ①: y＝2×2－1＝3
x＝2, y＝3
{y＝2x－1 … ①
{3x＋y＝9 … ②
Solution:
Substitute ① into ②: 3x＋(2x－1)＝9
5x－1＝9
5x＝10 → x＝2
Substitute into ①: y＝2×2－1＝3
x＝2, y＝3
Example 2
{2x＋y＝7 … ①
{x－3y＝－4 … ②
Solution:
From ①, y＝7－2x … ①'
Substitute into ②: x－3(7－2x)＝－4
x－21＋6x＝－4
7x＝17 → x＝17/7
※If the result does not divide evenly, try the elimination method
{2x＋y＝7 … ①
{x－3y＝－4 … ②
Solution:
From ①, y＝7－2x … ①'
Substitute into ②: x－3(7－2x)＝－4
x－21＋6x＝－4
7x＝17 → x＝17/7
※If the result does not divide evenly, try the elimination method
Key Points
- The substitution method is convenient when there is an equation in the form y＝ or x＝
- After substitution, be sure to expand the parentheses correctly
- The elimination method and substitution method give the same answer. Choose whichever makes the calculations easier
Practice Problems
- {y＝3x＋1 {2x＋y＝11 Solve using the substitution method.
- {x＝y＋4 {3x－2y＝10 Solve using the substitution method.
Answers and Explanations
- Answer:x＝2, y＝7
Explanation: ①y＝3x＋1, ②2x＋y＝11. Substitute ① into ②: 2x＋(3x＋1)＝11. 5x＝10. x＝2. y＝3×2＋1＝7. Check: 2×2＋7＝11 ✓ - Answer:x＝2, y＝−2
Explanation: ①x＝y＋4, ②3x－2y＝10. Substitute ① into ②: 3(y＋4)－2y＝10. 3y＋12－2y＝10. y＝－2. x＝－2＋4＝2. Check: 3×2－2×(－2)＝6＋4＝10 ✓