GCSE Maths Algebra - How to Solve: Linear Equations and Inequalities

How to Solve: Linear Equations and Inequalities

GCSE / A-Level (Physics, Chemistry, Biology) – Complete Guide

Introduction

"Mastering linear equations and inequalities lets you solve real-world problems—like calculating drug dosages in Biology, reaction rates in Chemistry, or motion equations in Physics—and could earn you 10-15% of your exam marks in algebra-based questions."

WHAT YOU NEED TO KNOW FIRST

1. Basic arithmetic (addition, subtraction, multiplication, division).
2. Balancing equations (whatever you do to one side, you must do to the other).
3. Negative numbers (how to multiply/divide them correctly).

KEY TERMS & FORMULAS

Key Terms

• Linear equation: An equation where the highest power of the variable is 1 (e.g., 2x + 3 = 7).
• Inequality: A statement that compares two expressions (e.g., x + 5 > 10).
• Solution set: All possible values of the variable that make the equation/inequality true.
• Coefficient: The number multiplied by the variable (e.g., in 3x, 3 is the coefficient).

Formulas

1. Standard form of a linear equation:
   ax + b = c
2. a = coefficient of x
3. b = constant term
4. c = constant on the other side

5. MEMORISE THIS (used in every linear equation problem).

6. Inequality symbols:

7. > (greater than)
8. < (less than)
9. ≥ (greater than or equal to)
10. ≤ (less than or equal to)

11. MEMORISE THESE (examiners test if you mix them up).

12. Multiplying/dividing by a negative number in inequalities:

13. Rule: If you multiply or divide both sides by a negative number, flip the inequality sign.
14. MEMORISE THIS (most common mistake in inequalities).

STEP-BY-STEP METHOD

Solving Linear Equations (Step-by-Step)

1. Simplify both sides (remove brackets, combine like terms).
2. Move all x terms to one side (use addition/subtraction).
3. Move all constants to the other side (use addition/subtraction).
4. Divide by the coefficient of x to isolate x.
5. Check your answer by substituting back into the original equation.

Solving Linear Inequalities (Step-by-Step)

1. Follow the same steps as equations (simplify, move terms, divide).
2. If multiplying/dividing by a negative number, flip the inequality sign.
3. Write the solution in inequality form (e.g., x > 3).
4. Draw a number line (if required) to show the solution set.
5. Check a value from your solution to ensure it works.

WORKED EXAMPLES

Example 1 – Basic Linear Equation

Problem: Solve 3x + 5 = 14.

Step-by-Step Solution: 1. Subtract 5 from both sides:
3x + 5 – 5 = 14 – 5
3x = 9 2. Divide both sides by 3:
3x ÷ 3 = 9 ÷ 3
x = 3 3. Check: Substitute x = 3 into the original equation:
3(3) + 5 = 9 + 5 = 14 ✔️

What we did and why: - We isolated x by first removing the constant (+5) and then dividing by the coefficient (3). - Always check your answer to avoid mistakes.

Example 2 – Medium Linear Equation (with brackets)

Problem: Solve 2(x – 4) + 3 = 7.

Step-by-Step Solution: 1. Expand the bracket:
2x – 8 + 3 = 7 2. Combine like terms:
2x – 5 = 7 3. Add 5 to both sides:
2x = 12 4. Divide by 2:
x = 6 5. Check: Substitute x = 6:
2(6 – 4) + 3 = 2(2) + 3 = 4 + 3 = 7 ✔️

What we did and why: - We expanded the bracket first to simplify the equation. - Combining like terms (-8 + 3) made the equation easier to solve.

Example 3 – Exam-Style Inequality

Problem: Solve 5 – 2x ≤ 11 and represent the solution on a number line.

Step-by-Step Solution: 1. Subtract 5 from both sides:
-2x ≤ 6 2. Divide by -2 (remember to flip the inequality sign):
x ≥ -3 3. Number line representation:
- Draw a closed circle at -3 (because x can equal -3).
- Shade the line to the right (because x is greater than -3).

What we did and why: - We flipped the inequality sign when dividing by a negative number. - A closed circle means -3 is included in the solution.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP (Night Before the Exam)

"Okay, listen up—this is your last-minute crash course for linear equations and inequalities. First, equations: simplify, move x terms to one side, constants to the other, then divide. Always check your answer by plugging it back in. For inequalities, do the same steps, but flip the sign if you multiply or divide by a negative. Number lines? Closed circle for ≥ or ≤, open for > or <. Watch out for traps: brackets, negative coefficients, and word problems. If you see ‘at least,’ that’s ≥. ‘No more than’ is ≤. You’ve got this—go ace that exam!"