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Mastering Linear Equations

CSEC Mathematics: Algebraic Solutions

Essential Understanding: Linear equations are the foundation of algebraic problem-solving. Understanding how to solve them allows you to find unknown values, model real-world situations, and build skills for more advanced mathematics.

🔑 Key Skill: Inverse operations

📈 Exam Focus: Multi-step solutions

🎯 Problem Solving: Word problems

Core Concepts: Understanding Equations

Linear Equation

Definition: A mathematical statement that shows two expressions are equal, containing one or more variables raised to the power of 1.

Standard Form: $ ax + b = c $ where $ a, b, c $ are constants and $ a \neq 0 $

Examples: $ 3x = 12 $, $ 2x + 5 = 13 $, $ x – 7 = -3 $

Key Point: The graph of a linear equation is always a straight line!

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Solution (Root)

Definition: The value of the variable that makes the equation true.

Example: For $ x + 5 = 12 $, the solution is $ x = 7 $ because $ 7 + 5 = 12 $.

Verification: Always substitute your answer back into the original equation to check!

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Inverse Operations

Definition: Operations that “undo” each other. Used to isolate the variable.

Addition ↔ Subtraction

Multiplication ↔ Division

Key Point: Whatever operation you do to one side, you MUST do to the other side!

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The Balance Method

Concept: Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced.

Rule: The Golden Rule – “Do unto one side as you do unto the other!”

Goal: Get the variable alone on one side of the equation.

The Balance Method Visualized

One-Step Equations

The Four Types of One-Step Equations

Solve by applying the inverse operation to isolate x:

Addition Equation

$ x + a = b $

Inverse: Subtract a

$ x = b – a $

Subtraction Equation

$ x – a = b $

Inverse: Add a

$ x = b + a $

Multiplication Equation

$ ax = b $

Inverse: Divide by a

$ x = b \div a $

Division Equation

$ \frac{x}{a} = b $

Inverse: Multiply by a

$ x = b \times a $

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Worked Example 1: Addition Equation

Solve: $ x + 7 = 15 $

Identify the operation: 7 is being added to x, so we subtract 7 from both sides.

Apply inverse operation:
$ x + 7 – 7 = 15 – 7 $

Simplify:
$ x = 8 $

Check: $ 8 + 7 = 15 $ ✓
Answer: x = 8

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Worked Example 2: Division Equation

Solve: $ \frac{x}{4} = 6 $

Identify the operation: x is divided by 4, so we multiply both sides by 4.

Apply inverse operation:
$ \frac{x}{4} \times 4 = 6 \times 4 $

Simplify:
$ x = 24 $

Check: $ \frac{24}{4} = 6 $ ✓
Answer: x = 24

Two-Step Equations

⚠️ Order Matters!

When solving equations with multiple operations, follow the reverse order of operations (BODMAS backwards!):

Step 1: Add or subtract first (undo addition/subtraction)

Step 2: Then multiply or divide (undo multiplication/division)

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Worked Example 3: Two-Step Equation

Solve: $ 3x + 5 = 17 $

Identify the operations: First, x is multiplied by 3. Then 5 is added.

Undo the addition first:
$ 3x + 5 – 5 = 17 – 5 $
$ 3x = 12 $

Undo the multiplication:
$ \frac{3x}{3} = \frac{12}{3} $
$ x = 4 $

Check: $ 3(4) + 5 = 12 + 5 = 17 $ ✓
Answer: x = 4

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Worked Example 4: Two-Step with Subtraction

Solve: $ 2x – 7 = 9 $

Undo the subtraction:
$ 2x – 7 + 7 = 9 + 7 $
$ 2x = 16 $

Undo the multiplication:
$ \frac{2x}{2} = \frac{16}{2} $
$ x = 8 $

Check: $ 2(8) – 7 = 16 – 7 = 9 $ ✓
Answer: x = 8

Equations with Variables on Both Sides

Strategy: Collect Variables on One Side

When variables appear on both sides of the equation:

Step 1: Move all variable terms to one side (usually left)

Step 2: Move all constant terms to the other side

Step 3: Simplify and solve

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Worked Example 5: Variables on Both Sides

Solve: $ 4x + 6 = 2x + 14 $

Move 2x to the left side:
Subtract 2x from both sides:
$ 4x – 2x + 6 = 2x – 2x + 14 $
$ 2x + 6 = 14 $

Move constant to the right:
Subtract 6 from both sides:
$ 2x + 6 – 6 = 14 – 6 $
$ 2x = 8 $

Divide by 2:
$ \frac{2x}{2} = \frac{8}{2} $
$ x = 4 $

Check: $ 4(4) + 6 = 16 + 6 = 22 $ and $ 2(4) + 14 = 8 + 14 = 22 $ ✓
Answer: x = 4

Interactive Equation Solver

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Equation Solving Practice

Objective: Solve linear equations by applying inverse operations step by step.

Choose an Equation Type

Current Equation

Select an equation to solve

Step 1: Inverse Operation

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Step 2: Simplify

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Final Answer

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Fraction Equations

⚠️ Clearing Fractions

When equations contain fractions, multiply EVERY term by the least common denominator (LCD) to eliminate fractions!

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Worked Example 6: Equation with Fractions

Solve: $ \frac{x}{3} + 2 = 8 $

Identify the LCD: The denominator is 3, so LCD = 3.

Multiply every term by 3:
$ 3 \times \frac{x}{3} + 3 \times 2 = 3 \times 8 $
$ x + 6 = 24 $

Solve the resulting equation:
$ x = 24 – 6 $
$ x = 18 $

Check: $ \frac{18}{3} + 2 = 6 + 2 = 8 $ ✓
Answer: x = 18

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Worked Example 7: Complex Fraction

Solve: $ \frac{2x + 4}{5} = 6 $

Multiply every term by 5:
$ 5 \times \frac{2x + 4}{5} = 5 \times 6 $
$ 2x + 4 = 30 $

Subtract 4 from both sides:
$ 2x = 26 $

Divide by 2:
$ x = 13 $

Check: $ \frac{2(13) + 4}{5} = \frac{26 + 4}{5} = \frac{30}{5} = 6 $ ✓
Answer: x = 13

Word Problems Leading to Linear Equations

The Word Problem Strategy

Step 1: Read the problem carefully and identify what you’re solving for

Step 2: Assign a variable to the unknown quantity

Step 3: Translate the words into a mathematical equation

Step 4: Solve the equation

Step 5: Check your answer and write it in context

Word Problem Example 1: Number Problems

Problem: Five times a number plus 3 equals 28. Find the number.

Define the variable:
Let $ x $ = the number

Write the equation:
“Five times a number” = $ 5x $
“Plus 3” = $ + 3 $
“Equals 28” = $ = 28 $
So: $ 5x + 3 = 28 $

Solve:
$ 5x = 28 – 3 $
$ 5x = 25 $
$ x = 5 $

Answer: The number is 5

Word Problem Example 2: Age Problems

Problem: Maria is twice as old as her brother. The sum of their ages is 24. How old is Maria?

Define the variables:
Let $ b $ = brother’s age
Then Maria’s age = $ 2b $ (twice as old)

Write the equation:
“Sum of their ages” = $ b + 2b $
“Is 24” = $ = 24 $
So: $ b + 2b = 24 $

Solve:
$ 3b = 24 $
$ b = 8 $ (brother’s age)

Find Maria’s age:
Maria = $ 2 \times 8 = 16 $
Answer: Maria is 16 years old

Word Problem Example 3: Money Problems

Problem: A shirt costs $15 more than a pair of shorts. Together they cost $75. Find the cost of the shorts.

Define the variable:
Let $ s $ = cost of shorts ($)
Then shirt = $ s + 15 $ ($15 more)

Write the equation:
Cost together = shorts + shirt
$ s + (s + 15) = 75 $

Solve:
$ 2s + 15 = 75 $
$ 2s = 60 $
$ s = 30 $

Answer: The shorts cost $30 (Shirt = $45, total = $75) ✓

Word Problem Example 4: Distance/Speed Problems

Problem: A car travels at a constant speed for 3 hours. If it travels a total of 180 km, what is its speed in km/h?

Recall the formula:
Distance = Speed × Time
So: $ d = v \times t $

Define the variable and substitute:
Let $ v $ = speed (km/h)
$ 180 = v \times 3 $

Solve:
$ v = \frac{180}{3} $
$ v = 60 $

Answer: The speed is 60 km/h

Special Cases: No Solution or Infinite Solutions

⚠️ Watch Out for These Cases!

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No Solution

When: After simplifying, you get a false statement like $ 5 = 7 $

Example: $ 2x + 3 = 2x + 7 $

Result: Subtract 2x: $ 3 = 7 $ (Impossible!)

Answer: No solution

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Infinite Solutions

When: After simplifying, you get a true statement like $ 7 = 7 $

Example: $ 3x + 5 = 3x + 5 $

Result: Subtract 3x: $ 5 = 5 $ (Always true!)

Answer: All real numbers

Key Examination Insights

Common Mistakes to Avoid

Forgetting to do the same operation to BOTH sides
Doing operations in the wrong order (not reverse BODMAS)
Forgetting to check answers by substitution
Making sign errors, especially with negative numbers
Not clearing fractions before solving
Leaving the equation unbalanced

Success Strategies

Always write each step clearly – show your work!
Isolate the variable step by step
Check every answer by substituting back
For word problems, always state what the variable represents
Write the final answer in context for word problems
Practice all types of equations regularly

CSEC Practice Arena

Test Your Understanding

Solve: $ x + 8 = 20 $

x = 12

x = 28

x = 16

Solution: Subtract 8 from both sides: x = 20 – 8 = 12.

Solve: $ 5x = 35 $

x = 7

x = 30

x = 7

x = 40

Solution: Divide both sides by 5: x = 35 ÷ 5 = 7.

Solve: $ 2x + 7 = 15 $

x = 4

x = 11

x = 5.5

Solution: Subtract 7: 2x = 8, then divide by 2: x = 4.

Solve: $ 3x – 4 = 2x + 6 $

x = 10

x = 2

x = 8

Solution: Subtract 2x: x – 4 = 6, then add 4: x = 10.

Solve: $ \frac{x}{4} – 2 = 5 $

x = 12

x = 28

x = 3

Solution: Add 2: x/4 = 7, then multiply by 4: x = 28.

Three times a number minus 5 equals 16. Find the number.

Solution: Equation: 3x – 5 = 16, so 3x = 21, x = 7.

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CSEC Examination Mastery Tip

Past Paper Patterns: Linear equations appear throughout CSEC Mathematics papers. Common question types include:

Direct solving: Solve equations of varying complexity
Word problems: Translate real scenarios into equations
Formula rearrangement: Change the subject of formulas
Verification: Show that a value satisfies an equation

Tip: Always show your working step by step. CSEC examiners award method marks even if your final answer has a small error. Writing each transformation clearly demonstrates your understanding!

Extended Practice Questions

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CSEC-Style Question 1

(a) Solve: $ 4x + 9 = 21 $

(b) Solve: $ 7x – 3 = 3x + 13 $

(c) Solve: $ \frac{x + 5}{2} = 8 $

Answer Key:
(a) 4x = 12, so x = 3
(b) 4x = 16, so x = 4
(c) x + 5 = 16, so x = 11

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CSEC-Style Question 2

(a) The perimeter of a rectangle is 48 cm. If the length is twice the width, find the length.

(b) A number is increased by 12 and then doubled. The result is 50. Find the original number.

(c) Joan has some stickers. After giving 5 to her friend, she has 17 left. How many stickers did Joan have originally?

Answer Key:
(a) Let w = width, then length = 2w. Perimeter = 2(w + 2w) = 6w = 48, so w = 8 cm, length = 16 cm
(b) Let x = number. 2(x + 12) = 50, x + 12 = 25, x = 13
(c) Let s = stickers. s – 5 = 17, s = 22 stickers

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CSEC-Style Question 3

(a) Solve: $ 2(x + 3) = 5(x – 1) $

(b) Solve: $ 3x – 7 = 2(x + 4) + x $

(c) The difference between two numbers is 8. If the larger number is three times the smaller number, find both numbers.

Answer Key:
(a) 2x + 6 = 5x – 5, 11 = 3x, x = 11/3
(b) 3x – 7 = 2x + 8 + x, 3x – 7 = 3x + 8, -7 = 8, No solution
(c) Let smaller = x, larger = x + 8. x + 8 = 3x, 8 = 2x, x = 4. Numbers are 4 and 12

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CSEC-Style Question 4

(a) Change the subject to p: $ I = \frac{PRT}{100} $

(b) Change the subject to r: $ A = P(1 + r) $

(c) Change the subject to h: $ V = \frac{1}{3}\pi r^2 h $

Answer Key:
(a) $ P = \frac{100I}{RT} $
(b) $ r = \frac{A}{P} – 1 $ or $ r = \frac{A – P}{P} $
(c) $ h = \frac{3V}{\pi r^2} $