Solving Two-Step Equations
Solve one-variable linear equations with natural number solutions; verify the solution by substitution.
An equation is a balance. What you do to one side, you do to the other.
Builds onYr 7 · Order of operationsYr 6 · Inverse operations
Why this works
Picture an equation as a set of scales that balance.
says: three unknown weights plus 5 grams balance against 20 grams.
Remove 5 grams from the left and the scales tip — unless you remove 5 from the right as well. Balance is preserved only when both sides are treated identically.
That leaves three weights balancing 15 grams.
Three equal weights balance 15, so one weight is 5. Dividing both sides by 3 keeps the balance.
The order also matters, and it is the reverse of BODMAS. Building from means multiply then add; undoing it means subtract then divide. Unwrap in the opposite order to wrapping.
Worked example
e.g.
- Identify what has been done to : multiplied by 4, then 7 subtracted.
- Undo the last operation first. The opposite of subtracting 7 is adding 7 — to both sides.
- Simplify.
- Now undo the multiplication. Divide both sides by 4.
- Simplify to get the solution.
- Check your answer. Put 7 back into the first equation.
Common mistakes
Wrong
Right
Adding 5 instead of subtracting it. To undo you must do the opposite. Ask yourself each time: what is the inverse?
Wrong
Right
Undo in reverse BODMAS order. Dividing by 3 first gives — not wrong, but far messier than necessary.
Wrong
Right
Subtracting 4 instead of dividing. means , so the inverse is division, not subtraction.
Wrong
Right
Substitution takes ten seconds and catches almost every error. Your exam names it as part of the skill, so it is assessable in its own right.
Practice
FluencyGet quick and accurate at the method.
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ReasoningExplain why. Say it in your own words.
1
Why must you subtract before dividing when solving ?
AnswerBecause you undo operations in the reverse of the order they were applied. Building the expression multiplied first and added second, so unwrapping subtracts first and divides second. Dividing first still works but drags fractions through the whole calculation.
2
Sam solves by subtracting 4 from both sides. Explain his error using the balance idea.
Answer means four lots of , not plus . Removing 4 grams does not remove three of the four weights. To get from four weights to one you must divide the whole side by 4.
3
How does substituting your answer back in prove you are right?
AnswerThe equation claims both sides are equal for the correct value of . If substituting makes the two sides genuinely equal, the value satisfies the equation. If they differ, there is an error somewhere in the working.
4
Would and have the same solution? Explain without solving either.
AnswerYes. Expanding gives exactly , so the two equations say the same thing written differently. Equivalent expressions always give equivalent equations.
AppliedThe same maths, inside a real question.
1
A plumber charges a call-out fee plus per hour. The bill is . How many hours did the job take?
Answer4 hours. , so and . Check: ✓
2
Mia buys 5 identical notebooks and a pen, spending in total. What does one notebook cost?
Answer. , so and .
3
A rectangle has a perimeter of 34 cm and a width of 6 cm. Find its length.
Answer11 cm. , so , giving and .
4
A phone plan costs a month plus per text. Kai's bill is . How many texts did he send?
Answer60 texts. , so and .