Solve linear equations with whole-number coefficients

What you will learn

• Solve linear equations with whole-number coefficients

Models used: number sentences, stories, diagrams.

Before you start

• Letters and balanced equations

Connect the action to the maths

Two identical mystery bags and three counters balance eleven counters. How many counters must be in each bag?

1. Matching bags hold the same amount. A level balance means both sides have equal value.

   Two identical bags and three loose counters have the same total as eleven counters. The letter x stands for the unknown number in each bag.

   2x + 3 = 11

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xx

   2x + 3 = 11 · balanced

2. Matching bags hold the same amount. A level balance means both sides have equal value.

   Remove three loose counters from each side. Both totals shrink by three, so the scale remains level.

   2x = 8

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xx

   2x = 8 · balanced

3. Matching bags hold the same amount. A level balance means both sides have equal value.

   Share both sides into two equal groups. One bag has the same value as four counters.

   x = 4

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   x

   x = 4 · balanced

4. Matching bags hold the same amount. A level balance means both sides have equal value.

   Check the starting situation, not just the final line. Two bags of four plus three loose counters make eleven.

   2 × 4 + 3 = 11

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xx

   2x + 3 = 11 · balanced

Worked example: Try a new balanced equation

Solve 3x + 2 = 14.

1. Matching bags hold the same amount. A level balance means both sides have equal value.

   Three equal bags and two loose counters balance fourteen. First remove two from both sides.

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xxx

   3x + 2 = 14 · balanced

   What total remains on the right after subtracting two?

   12

2. Matching bags hold the same amount. A level balance means both sides have equal value.

   Three bags now balance twelve counters. Share the twelve equally among the three bags.

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xxx

   3x = 12 · balanced

   Solve 3x + 2 = 14.

   4

3. Matching bags hold the same amount. A level balance means both sides have equal value.

   Put four into the original equation. Three groups of four plus two is fourteen, so both sides match.

   3 × 4 + 2 = 14

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xxx

   3x + 2 = 14 · balanced

Common mistakes to discuss

• Changing one side only. An equation stays true when both sides change together. From 2x + 3 = 11, take 3 from each side, then halve each side.

Check your understanding

• Solve an equation by making equal changes to both sides, then check the unknown in the original equation.
• explain why an equation-solving step preserves equality
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.