Year 9 · Unit 7 Equations and linear systems · Lesson 1 of 7

Keeping the balance when brackets get in the way

An equation balances at one value of the unknown. Expand the brackets, gather the unknowns on one side, and undo what is left, doing the same to both sides each time.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Solve 3x + 4 = 19.
    Answerx = 5, because 19 − 4 = 15 and 15 ÷ 3 = 5.
  2. Simplify 5x + 3 − 2x + 7.
    Answer3x + 10: collect the x terms (5x − 2x) and the numbers (3 + 7).
  3. Work out −4 × (−3).
    Answer12: a negative times a negative is positive.
Words used in this lesson
equation
a statement that two expressions are equal. For example: 3x + 4 = 19.
expand
multiply out a bracket. For example: 3(x + 5) expands to 3x + 15.
linear
the unknown is only to the power 1. For example: 5x − 2 = 3x + 6.
degree
the highest power of the unknown. For example: x² + 3x = 10 has degree 2.
solution
the value that makes both sides equal. For example: x = 3 solves 2x + 1 = 7.
inverse operation
the operation that undoes another. For example: subtract 3 undoes add 3.

Same total, two ways of packing

A fair-trade tea stall in Nairobi packs 4 boxes. Each box holds x tea bags and 3 free samples, so the total is 4(x + 3). The stall's other table has 2 boxes of x tea bags and one bag of 18 loose samples: 2x + 18. Both tables hold the same total.

Predict first: is x = 1, x = 3 or x = 5 tea bags a box? Then slide the value of x and watch the two bars.

4(x + 3) = 2x + 18
1
16left side
20right side
right is 4 morethe balance

4(x + 3) = 2x + 18 at x = 1: left side 16, right side 20. Not balanced.

Show me the steps
  1. 4(x + 3) = 2x + 18 at x = 1: left 4 × 4 = 16, right 2 + 18 = 20. Not balanced; the right side is 4 more.
  2. The same equation at x = 3: left 4 × 6 = 24, right 6 + 18 = 24. Balanced, so x = 3.
  3. 3(x − 2) = x + 8 at x = 7: left 3 × 5 = 15, right 7 + 8 = 15. Balanced, so x = 7.
  4. 2(3x − 4) = 4(x + 1) at x = 6: left 2 × 14 = 28, right 4 × 7 = 28. Balanced, so x = 6.

An equation is a balance. The solution is the one value of the unknown that makes both sides equal. To find it without guessing, do the same thing to both sides until x stands alone:

1. Expand each bracket: multiply every term inside by the number outside.
2. Take the x terms from one side, so they are only on the other.
3. Take the plain numbers from the other side, so they are only where the x terms are not.
4. Divide by the number in front of x, then check by putting the answer back.

For the tea stall: 4x + 12 = 2x + 18, so 2x = 6 and x = 3 tea bags a box.

The degree of an equation

The degree of an equation is the highest power of the unknown, after expanding. Degree 1 is linear, degree 2 is quadratic, degree 3 is cubic and degree 4 is quartic. Everything in this unit is linear: x is never squared. An equation can look linear and not be: 2x(x − 1) = 12 expands to 2x² − 2x = 12, which is quadratic.

A common mistake: multiplying only the first term in the bracket, so 4(x + 3) becomes 4x + 3. Test it with x = 1: 4 × (1 + 3) = 16, but 4 × 1 + 3 = 7. The 4 multiplies everything inside: 4x + 12.

Worked example: brackets and unknowns on both sides

Solve 3(2x − 1) = 5x + 4.

  1. Expand: 6x − 3 = 5x + 4.
  2. Take 5x from both sides: x − 3 = 4.
  3. Add 3 to both sides: x = 7.
  4. Check: 3(14 − 1) = 39 and 5 × 7 + 4 = 39. x = 7.

Your go. The first step is given. Solve 2(x + 5) = 6x − 2.

  1. Expand: 2x + 10 = 6x − 2.
  2. Now gather the x terms and the numbers yourself, then check.
Check the answer

Take 2x from both sides: 10 = 4x − 2. Add 2: 12 = 4x. Divide by 4: x = 3. Check: 2 × 8 = 16 and 6 × 3 − 2 = 16.

Check before you go on

1. Which line is 3(x + 5) written without a bracket?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Solve 2(x + 4) = 18.

3. Which of these equations is linear?

Level 2 · Building

4. Solve 5(x − 2) = 3x + 6.

5. At a community garden each of 3 volunteers buys one seedling at x shillings and one pot at 15 shillings. The total is the same as 5 seedlings and 5 shillings of rope. What does one seedling cost, in shillings?

Level 3 · Stretch

6. Solve 4(2x − 3) = 3(x + 6) − 5.

7. Solve 9 − 2(x − 3) = x − 3.

Something to think about

Look at 2(x + 3) = 2x + 5. Expand the bracket and try to solve it. What happens, and what does it say about every value of x?

One way to think about it

2x + 6 = 2x + 5 leaves 6 = 5 once the 2x is taken from both sides, which is never true. No value of x balances it: the left side is always 1 more than the right. Equations can have no solution, and the next lessons ask why.

Exit ticket

8. Which line follows correctly from 5(2x − 3) = 4x + 9?