Algebra — Linear Equations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Linear Equations — Key Facts for NABE (Pakistan)

• Linear Equation: Highest power of variable is 1 → ax + b = 0
• Solution: x = -b/a (for equation ax + b = 0)
• System of 2 variables: Use substitution or elimination method
• Cross-multiplication for fractions: If a/b = c/d, then ad = bc
• ⚡ Exam tip: Always transpose terms correctly — move to other side changes sign

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Algebra — Linear Equations — NABE (Pakistan) Study Guide

Basic Concepts

Algebraic Expression: Numbers and variables combined with operations (3x + 5)

Algebraic Equation: An expression with an equal sign (3x + 5 = 14)

Linear Equation: Variable has power 1 (x² is NOT linear)

Single Variable Equations

Form: ax + b = 0

Solution: x = -b/a

Example 1: 3x + 12 = 0

• 3x = -12
• x = -4

Example 2: 5x - 15 = 25

• 5x = 25 + 15 = 40
• x = 8

Equations with Fractions

Method: Cross-multiplication or clear denominators

Example: (x/3) + (x/4) = 7

• LCM of 3 and 4 = 12
• Multiply both sides by 12: 4x + 3x = 84
• 7x = 84
• x = 12

Two-Variable Linear Equations

Standard Form: ax + by + c = 0

System of Two Equations:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

Methods of Solving

Method 1: Substitution

1. Express one variable in terms of the other from equation 1
2. Substitute into equation 2
3. Solve for the variable
4. Back-substitute to find the other variable

Method 2: Elimination

1. Multiply equations to make coefficients of one variable equal
2. Add or subtract equations to eliminate that variable
3. Solve for remaining variable
4. Substitute back to find the other variable

NABE Exam Pattern

Common question types:

1. Solve linear equations
2. Word problems leading to linear equations
3. System of two equations
4. Age-related problems
5. Mixture problems

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Algebra — Linear Equations — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Equation Fundamentals

Equation Types by Degree:

• Degree 0: Constant (5 = 5)
• Degree 1: Linear (3x + 7 = 0)
• Degree 2: Quadratic (x² + 5x + 6 = 0)
• Degree 3: Cubic (x³ + 2x² + 3x + 4 = 0)

Key Properties of Equations:

1. Adding same number to both sides doesn’t change solution
2. Subtracting same number from both sides doesn’t change solution
3. Multiplying both sides by same non-zero number doesn’t change solution
4. Dividing both sides by same non-zero number doesn’t change solution

2. Solving Linear Equations — Detailed

Step-by-Step Process:

1. Simplify both sides (remove parentheses, combine like terms)
2. Gather all variable terms on one side
3. Gather all constant terms on the other side
4. Isolate the variable
5. Check the solution

Example with Parentheses:

3(x + 2) - 2(x - 1) = 17
3x + 6 - 2x + 2 = 17    [distribute negative]
(x + 8) = 17
x = 9

Checking: 3(9+2) - 2(9-1) = 3(11) - 2(8) = 33 - 16 = 17 ✓

3. Cross-Multiplication

For proportion a/b = c/d:

ad = bc

Example: Solve (x+1)/3 = (x-2)/4

• 4(x+1) = 3(x-2)
• 4x + 4 = 3x - 6
• 4x - 3x = -6 - 4
• x = -10

4. System of Linear Equations — Elimination

Example:

2x + 3y = 16  ... (1)
3x + 2y = 14  ... (2)

Multiply (1) by 3 and (2) by 2:

6x + 9y = 48
6x + 4y = 28
Subtract: 5y = 20
y = 4

Substitute: 2x + 3(4) = 16 → 2x = 4 → x = 2

Verification: 3(2) + 2(4) = 6 + 8 = 14 ✓

5. System of Linear Equations — Substitution

Same System:

2x + 3y = 16
3x + 2y = 14

From (1): x = (16 - 3y)/2

Substitute in (2):

3[(16 - 3y)/2] + 2y = 14
(48 - 9y)/2 + 2y = 14
48 - 9y + 4y = 28
-5y = -20
y = 4

Then x = (16 - 12)/2 = 2

6. Determinant Method (Cramer’s Rule)

For system:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

x = (c₁b₂ - b₁c₂) / (a₁b₂ - b₁a₂)
y = (a₁c₂ - c₁a₂) / (a₁b₂ - b₁a₂)

Where denominator D = (a₁b₂ - b₁a₂)

If D = 0: Either no solution or infinite solutions

7. Word Problems — Setting Up Equations

Age Problems: Example: Father is 4 times as old as son. In 8 years, he will be 2.5 times as old. Find current ages.

Let son’s age = x, father’s age = 4x In 8 years: 4x + 8 = 2.5(x + 8) 4x + 8 = 2.5x + 20 1.5x = 12 x = 8

Son = 8 years, Father = 32 years

Work Problems (already covered in Time and Work)

Mixture Problems: Example: How much 20% acid solution to add to 50 ml of 60% acid to get 40% solution?

Let x = ml of 20% solution Total acid = 0.2x + 0.6(50) Total volume = x + 50 0.2x + 30 = 0.4(x + 50) 0.2x + 30 = 0.4x + 20 10 = 0.2x x = 50 ml

8. Infinite and No Solutions

Infinite Solutions (Same line):

2x + 4y = 10
4x + 8y = 20

Second equation = 2 × first equation (identical lines) Result: Infinitely many solutions

No Solution (Parallel lines):

2x + 4y = 10
2x + 4y = 14

Left sides are same but right sides different (parallel) Result: No solution

9. Common Mistakes to Avoid

1. Sign Errors: Be careful when transposing negative terms
2. Division by Zero: Never divide by zero
3. Distributing Negative: -2(x-3) = -2x + 6, not -2x - 6
4. Fractions: Clear denominators early to avoid mistakes
5. Checking: Always verify your solution

Practice Questions for NABE

1. Solve: 4(x - 3) - 2(x + 5) = 3(x + 1) - 7
2. Solve system: 5x + 3y = 7 and 3x - 5y = -23
3. A number is such that when 15 is subtracted from 3 times the number, the result is 45. Find the number.
4. The sum of two numbers is 45. One number is three times the other. Find the numbers.
5. Father is twice as old as son. Ten years ago, father was three times as old as son was then. Find their current ages.

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.