Fix the mistake: Simultaneous equations

Simultaneous Equations ① Elimination Method

Learn the “elimination method,” in which one variable is eliminated by adding or subtracting two equations.

Steps for the Elimination Method

  1. Make the absolute values of the coefficients of the variable you want to eliminate equal (multiply an equation by a suitable number if necessary)
  2. Add or subtract the two equations to eliminate one variable
  3. Find the value of the remaining variable
  4. Substitute the value found into one of the original equations to find the other variable
Example 1 (Coefficients are already equal)
{x＋y＝5 … ①
{x－y＝1 … ②

Solution:
①＋②: 2x＝6 → x＝3
Substitute into ①: 3＋y＝5 → y＝2
x＝3, y＝2
Example 2 (Making the coefficients equal)
{2x＋3y＝12 … ①
{3x－y＝7 … ②

Solution:
②×3: 9x－3y＝21 … ②'
①＋②': 11x＝33 → x＝3
Substitute into ②: 9－y＝7 → y＝2
x＝3, y＝2
Key Points
  • Make the absolute values of the coefficients of the variable you want to eliminate equal
  • If the signs of the coefficients are the same, subtract; if they are different, add
  • Substitute the value found into the “simpler equation”

Practice Problems

  1. {3x＋y＝11　{x－2y＝－2 Solve using the elimination method.
  2. {4x－3y＝5　{2x＋y＝7 Solve using the elimination method.

Answers and Explanations

  1. Answer:x＝20/7, y＝17/7
    Explanation: ①3x＋y＝11, ②x－2y＝－2. ①×2: 6x＋2y＝22. ①×2＋②: 7x＝20, so x＝20/7. Substitute into ① to get y＝11－60/7＝17/7. Check: 3×20/7＋17/7＝11, 20/7－2×17/7＝－2.
  2. Answer:x＝13/5, y＝9/5
    Explanation: ①4x－3y＝5, ②2x＋y＝7. ②×3: 6x＋3y＝21. ①＋②×3: 10x＝26, so x＝13/5. Substitute into ② to get y＝7－26/5＝9/5. Check: 4×13/5－3×9/5＝5, 2×13/5＋9/5＝7.

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