Lesson 4: Linear Equations. Pythagorean Theorem
Lesson 4: Linear Equations. Pythagorean Theorem
Solve linear equations and use the Pythagorean theorem.
In this lesson you will learn how to solve one step and multi step linear equations.
For example, x – 3 = 10 or 2h + 1 = – 9.
The lesson demonstrates how to solve linear equations with brackets.
For example, 2(x – 1) = 16.
Such equations are solved using inverse operations and reverse BEDMAS. The first step is always expanding the expression with brackets using the distributive multiplication property.
You will also discover the Pythagorean Theorem.
Want to see how the Pythagorean Theorem was invented?
Check out the 7 Cool Math Proofs blog post.
You will learn when and why to use it, how to set it up and solve the equation.
The challenge of using a Pythagorean Theorem is that it involves squares (multiplying the number by itself) and it is important to follow the BEDMAS and equation solving algorithm carefully in order to avoid mistakes. For example, the quantities should always be squared first, then added.
Pythagorean Theorem Short Animation
Key ideas: Linear Equations. Pythagorean Theorem
Solve linear equations with inverse operations, doing the same thing to both sides; expand brackets first.
In a right triangle, the Pythagorean theorem says a2 + b2 = c2, where c is the hypotenuse (the longest side, opposite the right angle) and a and b are the legs.
Worked examples
Example 1
Solve 4x − 9 = 15.
- Add 9: 4x = 24. Divide by 4: x = 6.
Example 2
Solve 2(x + 3) = 5x − 9.
- 2x + 6 = 5x − 9. Subtract 2x and add 9: 15 = 3x.
- x = 5. Check: 2(8) = 16 and 25 − 9 = 16 ✓
Example 3
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
- c2 = 62 + 82 = 36 + 64 = 100.
- c = 10 cm
Example 4
A 13 m ladder leans against a wall with its foot 5 m from the wall. How high up the wall does it reach?
- h2 + 52 = 132, so h2 = 169 − 25 = 144.
- h = 12 m
Common mistakes
- Adding the legs instead of their squares.
- Using the hypotenuse as a leg when finding a missing leg (subtract, do not add).
- Forgetting to take the square root at the end.
Questions students ask
Does the Pythagorean theorem work for every triangle?
No, only for right triangles.
Check your understanding
Practise with the Grade 8 quizzes. You’ll see right away if each answer is correct.