Year 9 Maths The Thai Guy

Lesson 8.1 · 25 min

Constructing and Solving Linear Equations

The balance method for equations with brackets and unknowns on both sides, and turning word problems into equations.

In this lesson you will learn to…

  • Solve an equation by doing the same thing to both sides
  • Solve equations with brackets and with the unknown on both sides
  • Construct an equation from a word problem or a diagram
  • Check a solution by substituting it back in

Why this matters

An equation is a question written in algebra: what number makes this true? Once you can turn a real situation into an equation and solve it, you can answer questions about ages, prices, perimeters and much more without guessing.

Two sides of a balance

Think of the equals sign as the middle of a set of scales. If x+3=8x + 3 = 8, adding 2 to both sides still balances: x+5=10x + 5 = 10. Subtracting 3 from both sides leaves x=5x = 5 — and the scales still balance.

Letters on both sides

Move the smaller letter term. In 6x−13=7−4x6x - 13 = 7 - 4x, add 4x4x to both sides rather than subtracting 6x6x — you avoid negative numbers:

10x−13=7  ⇒  10x=20  ⇒  x=210x - 13 = 7 \;\Rightarrow\; 10x = 20 \;\Rightarrow\; x = 2

Useful phrases

WordsAlgebra
three consecutive numbersx,  x+1,  x+2x,\; x + 1,\; x + 2
three consecutive odd (or even) numbersx,  x+2,  x+4x,\; x + 2,\; x + 4
20 more than xxx+20x + 20
twice as much as cc2c2c
3 is added, then multiplied by 44(x+3)4(x + 3)

Always check. Put your answer back into the original equation. For 7−3x=3(2x−5)7 - 3x = 3(2x - 5) with x=229x = \frac{22}{9}: left side =7−669=−13= 7 - \frac{66}{9} = -\frac{1}{3}, right side =3(449−5)=−13= 3\left(\frac{44}{9} - 5\right) = -\frac{1}{3} ✓

Key rules and formulas

  • Keep it balanced

    Whatever you do to one side of an equation, do exactly the same to the other. The aim is to get the letter on its own.

  • Inverse operations

    undoes and undoes . Remember means , so multiply by 7 to undo it.

  • A reliable order

    1 Expand any brackets. 2 Collect the letter terms on one side (add or subtract them). 3 Collect the numbers on the other side. 4 Divide by the number in front of the letter.

  • Building an equation

    Choose a letter for the unknown, write every other quantity in terms of it, then write down what they add up to (or equal).

Worked examples

Worked example 1

A fraction

Solve .

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Brackets and unknowns on both sides

Solve .

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Constructing an equation

The sum of three consecutive whole numbers is 60. What are they?

Try it yourself first, then reveal the solution one step at a time.

Practice: linear equations

10 questions

Ten questions. Where you type a solution, a fraction such as 22/9 is fine.
  1. Question 1 of 10Quick-fire drill
    Quick-fire: solve for the letter.

    10 questions · 75 seconds · answer as many as you can.

  2. Question 2 of 10Put in order
    Put the steps for solving in order.

    Put these in the right order. Drag them, or use the arrow buttons.

  3. Question 3 of 10Type the answer
    Solve .
  4. Question 4 of 10Step by step
    Solve step by step.
    Solve .
    1. Expand the left side.

  5. Question 5 of 10Type the answerChallenge
    Solve .
  6. Question 6 of 10Match the pairs
    Match each description to its equation.

    Pick an answer, then choose the row it belongs to. You can also drag it.

    • 3 is added to a number and the result is multiplied by 4 to give 32
    • A number is multiplied by 3, then 7 is added, and the result is doubled to give 80
    • Three consecutive numbers add up to 72
    • Three consecutive odd numbers add up to 69

  7. Question 7 of 10Type the answer
    Three consecutive odd numbers add up to 69. What is the smallest of them?
  8. Question 8 of 10Step by step
    Construct and solve an equation.
    One side of a rectangle is 4 cm longer than the other. The perimeter is 28 cm.
    1. Let the short side be . Write an expression for the perimeter.

  9. Question 9 of 10Type the answer
    Ram bought 5 cinema tickets. He paid with 20 dollars and got 75 cents change. How much did one ticket cost, in cents?
    cents
  10. Question 10 of 10Type the answer
    An isosceles triangle has two sides of cm and a base of 8 cm. Its perimeter is 32 cm. Find .
    cm