Simple Linear Equations: The Complete Secondary 1 Guide

Your foundation for all of secondary math algebra. Learn the balance method and master inverse operations to solve any linear equation.

Whatever you do to one side, you must do to the other!

What is a Linear Equation?

A linear equation is an equation where the variable (usually $x$ ) appears only to the power of 1. There are no $x^2$ , $x^3$ , or other powers.

Linear Equations

$x + 5 = 12$
$3x = 18$
$2x + 7 = 15$
$\frac{x}{4} - 3 = 5$

NOT Linear Equations

$x^2 + 5 = 21$ (quadratic)
$\frac{1}{x} = 3$ (reciprocal)
$2^x = 8$ (exponential)
$\sqrt{x} = 4$ (square root)

💡 Goal of Solving

Isolate the variable — get $x$ by itself on one side of the equation to find its value.

The Balance Method: Your Key to Success

Think of an equation as a balanced scale. Both sides must always be equal, just like weights on a scale must balance.

The Golden Rule

Whatever you do to one side of the equation, you MUST do to the other side.

If you add 5 to the left side, add 5 to the right side. If you divide the left side by 3, divide the right side by 3. This keeps the equation balanced!

Visualizing the Balance

Here’s the equation $x + 9 = 20$ shown on a balance scale:

To isolate x, we subtract 9 from both sides. The scale stays balanced!

Inverse Operations: Undoing Math

Inverse operations are opposite operations that “undo” each other. To isolate a variable, use the inverse operation to remove numbers from its side.

If the variable has…	Use the inverse…	Example
Addition (+)	Subtraction (-)	$x + 7 = 12 \rightarrow x = 12 - 7$
Subtraction (-)	Addition (+)	$x - 5 = 8 \rightarrow x = 8 + 5$
Multiplication (x)	Division (/)	$4x = 20 \rightarrow x = 20 \div 4$
Division (/)	Multiplication (x)	$\frac{x}{3} = 6 \rightarrow x = 6 \times 3$

💡 Memory Tip

Think of inverse operations as “doing the opposite to cancel out.” Addition and subtraction cancel each other. Multiplication and division cancel each other.

One-Step Equations

One-step equations require just one operation to solve. Perfect for building your foundation!

Example 1: Solving Addition Equations

Problem:

Solve for x: $x + 9 = 20$

Solution:

Step 1: Identify the operation on x. Here, 9 is added to x.
Step 2: Use the inverse operation. Subtract 9 from both sides.
$x + 9 - 9 = 20 - 9$
Step 3: Simplify: $x = 11$

Answer: x = 11

Example 2: Solving Multiplication Equations

Problem:

Solve for y: $4y = 24$

Solution:

Step 1: Identify the operation on y. Here, y is multiplied by 4.
Step 2: Use the inverse operation. Divide both sides by 4.
$\frac{4y}{4} = \frac{24}{4}$
Step 3: Simplify: $y = 6$

Answer: y = 6

Example 3: Solving Division Equations

Problem:

Solve for n: $\frac{n}{5} = 7$

Solution:

Step 1: Identify the operation on n. Here, n is divided by 5.
Step 2: Use the inverse operation. Multiply both sides by 5.
$\frac{n}{5} \times 5 = 7 \times 5$
Step 3: Simplify: $n = 35$

Answer: n = 35

Two-Step Equations

Two-step equations require two operations to solve. The key is knowing which operation to undo first!

Order of Operations (Reversed!):

1. First: Undo addition or subtraction (remove the constant)
2. Then: Undo multiplication or division (remove the coefficient)

This is the reverse of BODMAS/PEMDAS — we work “backwards” to isolate the variable!

Example 4: Two-Step Equation (Addition & Multiplication)

Problem:

Solve for x: $3x + 5 = 20$

Solution:

Step 1: Remove the constant first. Subtract 5 from both sides.
$3x + 5 - 5 = 20 - 5$
$3x = 15$
Step 2: Remove the coefficient. Divide both sides by 3.
$\frac{3x}{3} = \frac{15}{3}$
$x = 5$
Check: Substitute x = 5 back into the original equation.
$3(5) + 5 = 15 + 5 = 20$ ✓

Answer: x = 5

Example 5: Two-Step Equation (Subtraction & Multiplication)

Problem:

Solve for m: $6m - 8 = 22$

Solution:

Step 1: Remove the constant first. Add 8 to both sides.
$6m - 8 + 8 = 22 + 8$
$6m = 30$
Step 2: Remove the coefficient. Divide both sides by 6.
$\frac{6m}{6} = \frac{30}{6}$
$m = 5$
Check: Substitute m = 5 back.
$6(5) - 8 = 30 - 8 = 22$ ✓

Answer: m = 5

Solving Word Problems

Word problems test your ability to translate real-world situations into equations. Follow these steps:

Step-by-Step Approach

1. Read the problem carefully
2. Define the variable (what is unknown?)
3. Write the equation
4. Solve the equation
5. Check your answer makes sense

Common Word Clues

”more than” → Addition
”less than” → Subtraction
”times” / “product of” → Multiplication
”shared equally” → Division
”is” / “equals” → = sign

Example 6: Word Problem

Problem:

A baker started the day with some flour. After adding 7 grams of sugar to the mix, the total weight of ingredients is 18 grams. How much flour did the baker start with?

Solution:

Step 1: Let $y$ = initial amount of flour (in grams)
Step 2: Write the equation: $y + 7 = 18$
Step 3: Solve by subtracting 7 from both sides:
$y + 7 - 7 = 18 - 7$
$y = 11$
Step 4: Check: 11 + 7 = 18 ✓

Answer: The baker started with 11 grams of flour.

Common Mistakes to Avoid

❌ Mistake 1: Only changing one side

Wrong: $x + 5 = 12 \rightarrow x = 12$ (forgot to subtract 5 from right side)

Correct: $x + 5 = 12 \rightarrow x = 12 - 5 = 7$

❌ Mistake 2: Wrong order in two-step equations

Wrong: For $3x + 5 = 20$ , dividing by 3 first gives $x + 5 = 6.67$ (incorrect!)

Correct: Subtract 5 first, then divide by 3.

❌ Mistake 3: Sign errors with negative numbers

Wrong: $x - 8 = 5 \rightarrow x = 5 - 8 = -3$

Correct: $x - 8 = 5 \rightarrow x = 5 + 8 = 13$ (add to undo subtraction!)

❌ Mistake 4: Forgetting to check your answer

Always substitute your answer back into the original equation to verify it works!

Quick Reference Summary

Equation Type	Example	Steps to Solve
Addition	$x + 5 = 12$	Subtract 5 from both sides
Subtraction	$x - 3 = 10$	Add 3 to both sides
Multiplication	$4x = 20$	Divide both sides by 4
Division	$\frac{x}{5} = 3$	Multiply both sides by 5
Two-Step	$2x + 7 = 15$	1. Subtract 7, 2. Divide by 2

Practice Tips for Success

Do This

Write out each step clearly
Always check your answer by substitution
Practice with different variable letters
Work through word problems regularly
Use the balance method visualization

Remember

Inverse operations undo each other
In two-step: constants first, coefficients second
Keep the equation balanced at all times
Negative numbers follow the same rules
Take your time — accuracy over speed!

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Simple Linear Equations: The Complete Secondary 1 Math Guide

Simple Linear Equations: The Complete Secondary 1 Guide

What is a Linear Equation?

Linear Equations

NOT Linear Equations

The Balance Method: Your Key to Success

The Golden Rule

Visualizing the Balance

Inverse Operations: Undoing Math

One-Step Equations

Two-Step Equations

Solving Word Problems

Step-by-Step Approach

Common Word Clues

Common Mistakes to Avoid

Quick Reference Summary

Practice Tips for Success

Do This

Remember

Ready to Master Linear Equations?

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