Lesson 30: Solving Equations Practice

§Text Passage

1Summary 重點摘要

Four basic types: x+a=b, x-a=b, ax=b, x÷a=b. Practice each type, learn to isolate x.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Type 1: x + a = bExample: x + 7 = 15. Since 7 is added to x, subtract 7 from both sides: on the left, +7 and -7 cancel, leaving x; on the right, 15 - 7 = 8, so x = 8. Check: 8 + 7 = 15 matches, so the solution is correct. Addition equations are solved with subtraction. Another example: x + 3.2 = 9 — subtract 3.2 from both sides, x = 9 - 3.2 = 5.8; the aim of moving terms is to leave x alone on one side and gather all the numbers on the other.
2. Block 2: Type 2: x - a = bExample: x - 5 = 11. Since 5 is subtracted from x, add 5 to both sides: on the left, -5 and +5 cancel, leaving x; on the right, 11 + 5 = 16, so x = 16. Check: 16 - 5 = 11 matches. Subtraction equations are solved with addition, the opposite of type one. Another example: x - 12 = 7.5 — add 12 to both sides, x = 7.5 + 12 = 19.5; check 19.5 - 12 = 7.5. Whole numbers or decimals, the idea of cancelling is exactly the same.
3. Block 3: Type 3: ax = bExample: 3x = 18, meaning x multiplied by 3. Divide both sides by 3: on the left, 3x ÷ 3 leaves x; on the right, 18 ÷ 3 = 6, so x = 6. Check: 3 × 6 = 18, both sides equal. Multiplication equations are solved with division, and 3x means 3 × x. Another example: 6x = 4.8 — divide both sides by 6, x = 4.8 ÷ 6 = 0.8; check 6 × 0.8 = 4.8. Whatever the coefficient is, divide by it — the goal is always to make x's coefficient 1.
4. Block 4: Type 4: x ÷ a = bExample: x ÷ 4 = 5, meaning x divided by 4. Multiply both sides by 4: on the left, ÷4 and ×4 cancel, leaving x; on the right, 5 × 4 = 20, so x = 20. Check: 20 ÷ 4 = 5 matches. Division equations are solved with multiplication. Another example: x ÷ 2.5 = 4 — multiply both sides by 2.5, x = 4 × 2.5 = 10; check 10 ÷ 2.5 = 4. All four types share one principle: always do the same thing to both sides.
5. Block 5: Decimals TooDecimals can appear in equations too, and the method does not change. Example: x + 2.5 = 6; subtract 2.5 from both sides, and the right becomes 6 - 2.5 = 3.5, so x = 3.5. Check: 3.5 + 2.5 = 6, correct. Line up the decimal points when working out the arithmetic; the rest is the same as whole-number equations. Another example: x - 1.8 = 4.2 — add 1.8 to both sides, x = 4.2 + 1.8 = 6; check 6 - 1.8 = 4.2. Decimal addition and subtraction slip easily, so column form is the safest.
6. Block 6: Fractions TooFractions can appear as well. Example: x + 12 = 34; subtract 12 from both sides, giving x = 34 - 12. Make the denominators the same first: 12 = 24, so x = 34 - 24 = 14. Check: 14 + 24 = 34, correct. With fractions, rewrite them over a common denominator before subtracting. Another example: x - 14 = 12 — add 14 to both sides, x = 12 + 14 = 24 + 14 = 34; check 34 - 14 = 24 = 12, matching the original equation.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Goal: isolate x
• x + a = b: subtract a
• x - a = b: add a
• a x = b: divide a
• x ÷ a = b: multiply a