Chapter Overview

Suppose a box and 3 kg together weigh 11 kg.

If the box weighs $x$ kg:

The number that makes this statement true is the unknown value.

1. What Is an Equation?

Example:

The left side is called the LHS and the right side is the RHS.

2. Trial and Error

For:

try different values.

If $x=5$:

Not correct.

If $x=6$:

So:

Trial and error is useful for simple equations, but a systematic method is faster.

3. Same Operation on Both Sides

4. Multiplication and Division Equations

5. Inverse Operations

6. Two-Step Equations

7. Unknown on Both Sides

8. Checking a Solution

Always substitute the answer back into the original equation.

For:

and $x=5$:

LHS = RHS, so the solution is correct.

9. Word Problems

Translate the situation into an equation.

10. Expression vs Equation

An expression such as:

has no equality sign. An equation states that two expressions are equal:

The equation asks which value of $x$ makes the equality true.

11. The Balance Principle

An equation behaves like a balanced scale. If the same operation is performed on both sides, equality is preserved.

If:

subtract 7 from both sides:

so:

12. Solving Two-step Equations

Work backward through the operations around the unknown.

13. Unknown on Both Sides

When the unknown appears on both sides, collect letter terms on one side and number terms on the other.

Example:

Subtract $3x$ from both sides:

Subtract 2:

Therefore:

14. Forming Equations from Words

A word problem becomes easier when the unknown is named first.

Example: “Three times a number increased by 5 is 29.”

Let the number be $x$.

Solve:

15. Checking and Interpreting Solutions

A solution is not complete until it is checked in the original equation. Substitution verifies both the algebra and the arithmetic.

In a word problem, also check whether the answer makes sense. An equation may be solved correctly but the interpretation may still be wrong if units or context are ignored.

16. Complete Equation Notes

Unknowns and Solutions

A letter such as $x$ represents the value to be found. A solution is a value that makes the equation true.

For:

$x=5$ is a solution because:

But $x=6$ is not a solution because $6+4\ne9$.

One-step Equations

Use the inverse operation.

Add 7 to both sides:

For:

divide both sides by 5:

For:

multiply both sides by 6:

Brackets in Equations

Sometimes simplify or expand before solving.

Example:

Divide both sides by 3:

so:

Alternatively expand first:

Both methods give the same result.

Equations with Unknown on Both Sides

Example:

Subtract $4x$ from both sides:

Add 5:

Therefore:

Word Problems Step by Step

1. Choose a letter for the unknown.
2. Translate the information into an equation.
3. Solve using balanced operations.
4. Check the solution.
5. Write the answer with the correct unit or meaning.

Common Translation Phrases

• a number increased by 8 → $x+8$
• 8 less than a number → $x-8$
• 8 less than twice a number → $2x-8$
• three consecutive whole numbers → $x,x+1,x+2$
• twice a number is 14 → $2x=14$

Final Check

Always substitute the answer into the original equation. If the two sides are unequal, revisit your algebra or arithmetic.

17. More Equation Practice Notes

Equation-solving Checklist

Before finishing a solution, ask:

1. Did I perform the same operation on both sides?
2. Did I simplify correctly?
3. Did I keep signs correct?
4. Did I substitute the answer back into the original equation?
5. In a word problem, did I answer what was actually asked?

Consecutive-number Problems

If one whole number is $n$, the next is $n+1$ and the next is $n+2$.

If three consecutive numbers have sum 45:

So the numbers are 14, 15 and 16.

Common Mistakes

Quick Revision

Practice Questions

Apply Your Learning

Create an equation for each:

1. A number increased by 8 is 21.
2. Four times a number is 36.
3. Twice a number plus 5 is 17.

Solve each equation and check your answer by substitution.

Additional Practice

Chapter Summary