Simple linear equations

1Summary 重點摘要

An equation is a balance: do the same thing to both sides. Use inverse operations to free x, then check by substituting back.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What an equation isAn equation is a mathematical sentence containing an unknown, such as x + 3 = 7: x is the unknown, the equals sign states that both sides have the same value, and the sentence reads 'x plus 3 makes 7'. Solving means finding the value of the unknown that makes the sentence true — here x = 4, since 4 + 3 is exactly 7. Picture the equation as a balance scale: the two sides start level, and whatever operation is performed must be performed on both sides equally; this balance idea underpins every technique that follows.
2. 2. Addition equations: solve by subtractingFor an equation like x + 3 = 7, the aim is to remove the + 3 and leave x alone: subtract 3 from both sides, giving x + 3 − 3 = 7 − 3, which simplifies to x = 4. The reason is that taking equal weights off both pans keeps a balance level; in the language of inverse operations, addition is undone by subtraction. Both sides must be changed together — subtracting on one side only tips the balance and produces a wrong answer, the most common beginner's fault.
3. 3. Subtraction equations: solve by addingFor x − 5 = 2, the left side carries − 5, so add 5 to both sides: x − 5 + 5 = 2 + 5, which simplifies to x = 7. Subtraction is undone by addition, the mirror of the previous case: a minus is cancelled with a plus, a plus with a minus — the inverse-operation principle runs through all linear equations. The worked example x + 7 = 12 belongs to this family: subtracting 7 from both sides gives x = 12 − 7 = 5, matching the example's answer.
4. 4. Multiplication equations: solve by dividingFor 3x = 12, the x has been multiplied by 3, so divide both sides by 3: 3x ÷ 3 = 12 ÷ 3, simplifying to x = 4. Multiplication is undone by division, and the right side must read 12 ÷ 3, not 12 × 3 — the most common careless slip in this type. The worked example 3x = 21 divides both sides by 3 to give x = 21 ÷ 3 = 7. Keep the pair in mind: see a multiply, divide both sides, and x appears.
5. 5. Division equations: solve by multiplyingFor x ÷ 2 = 5, the x has been divided by 2, so multiply both sides by 2: x ÷ 2 × 2 = 5 × 2, simplifying to x = 10. Division is undone by multiplication, the exact mirror of the multiplication case. The usual error is forgetting to multiply the right side, leaving x = 5, or changing only one side; the safeguard is simple — every time one side is changed, change the other immediately, as a matched pair. With this, all four basic types are complete: undo whatever is attached to x.
6. 6. CheckingAfter solving, substitute the answer back into the original equation — this is checking. For x + 3 = 7 with x = 4, substituting gives 4 + 3 = 7, both sides equal, so the answer is right. Checking exposes sign or operation slips made while moving terms at almost no cost, and in the worked example both checks, 5 + 7 = 12 and 3 × 7 = 21, pass before the answers are accepted. Writing the check inside the examination working is advisable: it confirms correctness and sometimes earns method marks.
7. 7. Two-step equationsA two-step equation such as 2x + 3 = 11 carries both a multiply and an add, and is solved in two moves: remove the constant first, then the coefficient. Step one: subtract 3 from both sides to get 2x = 8; step two: divide both sides by 2 to get x = 4. The order cannot be reversed — dividing first produces fractions and needless risk. The motto is 'addition and subtraction first, multiplication and division after', the reverse of the usual order of operations. Verify as always: 2 × 4 + 3 = 11 confirms the answer before you commit it.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Equation is like balance, both sides equal
• Add/subtract: both sides same
• Multiply/divide: both sides same
• Check: substitute back
• Two-step: first add/subtract, then multiply/divide