Fix the mistake: Solving Linear Equations in One Variable

Equations ① What Are Equations? Properties of Equality

An equation such as 2x + 3 = 9 contains a letter whose value has not yet been determined and is true only when a particular number is substituted for that letter. Such an expression is called an equation. The value of the letter that makes the equation true is called a solution, and finding the solution is called solving the equation.

How to Check a Solution of an Equation

To determine whether a number is a solution of an equation, substitute the number and check whether the left-hand side and right-hand side are equal.

  • Example: In the equation 2x + 3 = 9, when x = 3, left-hand side = 2×3+3 = 9 and right-hand side = 9. They are equal, so x = 3 is a solution.
  • When x = 2, left-hand side = 2×2+3 = 7 and right-hand side = 9. They are not equal, so x = 2 is not a solution.
Example ①
Which of the following is a solution of the equation 3x − 4 = 11?
A) x = 3　B) x = 5　C) x = 7
Answer: Substitute each value.
A) 3×3−4 = 5（≠11）　B) 3×5−4 = 11（＝11）　C) 3×7−4 = 17（≠11）
Therefore, the solution is B) x = 5.

Properties of Equality

The properties of equality are used when solving equations. Remember these four properties.
When A = B,
　① Adding the same number to both sides keeps them equal: A + C = B + C
　② Subtracting the same number from both sides keeps them equal: A − C = B − C
　③ Multiplying both sides by the same number keeps them equal: A × C = B × C
　④ Dividing both sides by the same nonzero number keeps them equal: A ÷ C = B ÷ C（C≠0）

Think of a balance scale. The balance is maintained even if the same weight is placed on both sides or both sides are divided in half.

Example ②
Solve the equation x + 5 = 12 using the properties of equality.
Answer: Subtract 5 from both sides (Property ②).
　x + 5 − 5 = 12 − 5
　x = 7
Example ③
Solve the equation 4x = 20 using the properties of equality.
Answer: Divide both sides by 4 (Property ④).
　4x ÷ 4 = 20 ÷ 4
　x = 5
Key Points
  • A solution of an equation is a value of the letter that makes left-hand side = right-hand side after substitution
  • To check a solution, substitute it and see whether both sides are equal
  • Properties of equality: adding, subtracting, multiplying, or dividing both sides by the same number (except 0) preserves equality
  • To solve an equation means to put it in the form “x = (number)”

Practice Problems

  1. Determine whether x = −2 is a solution of the equation 5x + 8 = −2.
  2. Solve using the properties of equality: x − 7 = 4
  3. Solve using the properties of equality: −3x = 15

Answers and Explanations

  1. Answer: It is a solution
    Explanation: Left-hand side = 5×(−2)+8 = −10+8 = −2. Right-hand side = −2. They are equal, so x = −2 is a solution.
  2. Answer: x = 11
    Explanation: Add 7 to both sides (Property ①). x − 7 + 7 = 4 + 7 → x = 11.
  3. Answer: x = −5
    Explanation: Divide both sides by −3 (Property ④). −3x ÷ (−3) = 15 ÷ (−3) → x = −5.

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