Learning Objectives
- Explain what an equation is.
- Understand what the equals sign (=) means.
- Solve simple linear equations.
- Check your answers.
- Avoid common mistakes.
Introduction
This lesson develops Linear Equations from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🎬 Keep both sides balanced
Apply the same operation to both sides.
🤔 Think About This...
Imagine you have a balance scale. Everything on the left weighs exactly the same as everything on the right.
If you remove one book from the left side, what must you do on the right side?
Exactly! You must remove one book from the right side too. Otherwise the scale will no longer balance.
📖 What is an Equation?
An equation is a mathematical sentence showing that two expressions have the same value.
The equals sign (=) means "is equal to." It does not mean "the answer comes next." It tells us that both sides have the same value.
🧠 The StudyNest Thinking Method
👉 What am I trying to leave on its own?
The answer is always:
Everything you do should help isolate the variable.
🔑 Inverse Operations
To isolate the variable, we use the opposite operation.
| Operation | Opposite Operation |
|---|---|
| Add | Subtract |
| Subtract | Add |
| Multiply | Divide |
| Divide | Multiply |
📝 Worked Examples
Example 1
Solve:
🔍 StudyNest Question: What are we trying to leave on its own?
We want to leave x by itself.
5 is being added. Undo it by subtracting 5 from BOTH sides.
Example 2
Solve:
Undo subtraction by adding 8 to both sides.
Example 3
Solve:
🔍 StudyNest Question: What operation is being done to x?
x is being multiplied by 4. Undo multiplication by dividing both sides by 4.
Example 4
Solve:
x is being divided by 5. Undo division by multiplying both sides by 5.
Example 5
Solve:
🔍 StudyNest Question: What must we undo first?
We undo the addition first. Subtract 4 from both sides.
Now divide both sides by 3.
Example 6
Solve:
Add 7 to both sides.
Divide both sides by 5.
🪜 The StudyNest Equation Shortcut
- Undo: Use the opposite operation.
- Balance: Do the same thing to both sides.
- Simplify: Continue until the variable is alone.
🕵️ Check Your Answer
A good maths detective always checks the evidence.
Suppose we solved:
and found:
Substitute 7 back into the original equation:
🌍 Real-Life Example
A student buys a notebook and a pen.
The pen costs $2, and the total cost is $8.
Let the notebook cost be x.
Subtract 2 from both sides.
⚠ Common Mistakes
Doing something to only one side of the equation.
An equation must remain balanced.
Changing signs without showing the operation.
It is safer to write the same operation on both sides.
Dividing only one term instead of the entire side.
Stopping before the variable is completely alone.
Practice Questions
Solve the following equations.
- x + 6 = 14
- x − 9 = 12
- 5x = 35
- x ÷ 4 = 6
- 2x + 3 = 15
- 4x − 5 = 19
- 7x + 2 = 30
- 3x − 8 = 13
✅ Show Answers
- x = 8
- x = 21
- x = 7
- x = 24
- x = 6
- x = 6
- x = 4
- x = 7
🧠 Did You Know?
⭐ What You Can Do Now
- ✅ I understand what an equation is.
- ✅ I know that both sides must remain balanced.
- ✅ I can use inverse operations.
- ✅ I can solve one-step and two-step equations.
- ✅ I can check my answer by substitution.
🎉 Well Done!
You can now solve simple linear equations. This is one of the most useful skills in Algebra.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
📈 Your Progress
Lesson 9 of 22
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.