How to Solve: Linear Equations in One Variable

How to Solve: Linear Equations in One Variable

Introduction

"Master linear equations in one variable, and you unlock 5–10 marks in every SSC, Bank, or Railway exam—questions that test your speed, accuracy, and problem-solving under pressure."

(On camera: Hold up a past paper with a highlighted linear equation question.) "This one question type appears in every section—Quantitative Aptitude, Reasoning, even Data Interpretation. If you solve it fast and error-free, you’re already ahead of 80% of test-takers."

What You Need To Know First

Before diving in, ensure you’re comfortable with: 1. Basic arithmetic operations (addition, subtraction, multiplication, division). 2. Balancing equations (whatever you do to one side, you must do to the other). 3. Simplifying expressions (combining like terms, expanding brackets).

(On camera: Point to a whiteboard with these 3 points.) "If any of these feel shaky, pause here and review them first. Linear equations are just these skills in action."

Key Vocabulary

(On camera: Read each term aloud, then ask:) "Which term do you find most confusing? Let’s clarify it now."

Formulas To Know

(For linear equations in one variable, you don’t need complex formulas—just these rules.)

1. General Form of a Linear Equation
   ax + b = 0
2. a = coefficient of x (must not be zero).
3. b = constant term.

4. MEMORISE THIS: This is the standard form you’ll see in exams.

5. Solution Formula
   x = –b/a

6. Only use this if the equation is in ax + b = 0 form.
7. MEMORISE THIS: It’s a shortcut for simple equations.

(On camera: Write both formulas on the board.) "You won’t always see ax + b = 0 directly, but if you rearrange any linear equation, it’ll fit this form. The solution is just –b/a."

Step-by-Step Method

(Follow these 5 steps for every linear equation, no matter how complex.)

1. Simplify Both Sides
2. Remove brackets (using distributive property: a(b + c) = ab + ac).

3. Combine like terms (e.g., 3x + 2x = 5x).

4. Move Variable Terms to One Side, Constants to the Other

5. Use addition/subtraction to "transpose" terms.

6. Golden Rule: Change the sign when moving across the equals sign.

7. Combine Like Terms Again

8. Simplify both sides until you have ax = b or x = number.

9. Solve for the Variable

10. If ax = b, divide both sides by a to get x = b/a.

11. If x = number, you’re done!

12. Verify Your Answer

13. Plug the value back into the original equation.
14. If both sides are equal, your solution is correct.

(On camera: Demonstrate each step with hand motions—e.g., "move this term to the left" with a sweeping gesture.)

Worked Example Using the Steps

Equation: 3(x + 2) – 4 = 2x + 5

1. Simplify Both Sides
2. Expand brackets: 3x + 6 – 4 = 2x + 5

3. Combine constants: 3x + 2 = 2x + 5

4. Move Variable Terms to One Side, Constants to the Other

5. Subtract 2x from both sides: 3x – 2x + 2 = 5 → x + 2 = 5

6. Subtract 2 from both sides: x = 5 – 2

7. Combine Like Terms

8. x = 3

9. Solve for the Variable

10. Already solved: x = 3

11. Verify

12. Plug x = 3 into original equation:
    3(3 + 2) – 4 = 2(3) + 5 → 3(5) – 4 = 6 + 5 → 15 – 4 = 11 → 11 = 11 ✓

(On camera: Write each step slowly, narrating aloud.) "See how we followed the exact same 5 steps? No shortcuts—just methodical work."

Worked Examples

Example 1 – Basic

Equation: 5x – 3 = 2x + 9

1. Subtract 2x from both sides: 3x – 3 = 9
2. Add 3 to both sides: 3x = 12
3. Divide by 3: x = 4

What we did and why: - Moved all x terms to the left and constants to the right. - Solved step-by-step to isolate x.

Example 2 – Medium

Equation: 2(3x – 1) + 4 = 5x + 7

1. Expand brackets: 6x – 2 + 4 = 5x + 7
2. Combine constants: 6x + 2 = 5x + 7
3. Subtract 5x: x + 2 = 7
4. Subtract 2: x = 5

What we did and why: - First simplified the left side by expanding and combining. - Then moved terms to isolate x.

Example 3 – Exam-Style

Question: If 4(x – 2) + 3 = 2(2x + 1), then x = ? (Options: A) 1, B) 2, C) 3, D) 4)

1. Expand both sides: 4x – 8 + 3 = 4x + 2
2. Combine constants: 4x – 5 = 4x + 2
3. Subtract 4x from both sides: –5 = 2
4. Wait! This is a contradiction—no solution exists.

What we did and why: - The equation simplifies to a false statement (–5 = 2), meaning there’s no solution. - Exam trap: The options might trick you into picking a number, but the answer is "no solution."

(On camera: Circle the contradiction and say:) "This is why verification is crucial! Always plug your answer back in."

Common Mistakes

(On camera: Hold up a red pen and say:) "These mistakes cost marks. Circle them in your notes and watch out for them in practice."

Exam Traps

(On camera: Show a past paper question with one of these traps.) "Examiners love these traps. Train yourself to spot them by practicing with mixed questions."

1-Minute Recap

"Okay, listen up. Linear equations in one variable are all about one thing: isolating x. Here’s how you do it, step by step: 1. Simplify both sides—expand brackets, combine like terms. 2. Move all x terms to one side, constants to the other. Flip the sign when you move anything across the equals sign. 3. Combine again if needed, then solve for x by dividing. 4. Verify by plugging your answer back in. If it doesn’t work, you made a mistake—go back!

Watch out for traps: No solution? Infinite solutions? Fractions? Don’t panic—just follow the steps. And if the equation looks weird, rewrite it in ax + b = 0 form first.

Pro tip: Time yourself. In exams, you’ve got 30–45 seconds per question. Practice until you can solve these in your sleep. You’ve got this!

(On camera: Point to the camera and smile.) "Now go solve 10 practice questions. See you in the top 1%!