KS3 · Maths

Solving two-step linear equations

To solve 2x + 4 = 10, you could clear the + 4 first, or halve everything first. Do the two routes end up in the same place? Try both.

Maths · Solving equations

Which first move would you make?

Take 2x + 4 = 10. The x has two things stuck to it: a × 2 and a + 4. Choose a first move, then a second move. Then go back and try the others.

First move → Second move

3 possible routes.

On 2x + 4 = 10. 3 branches to choose from.

The whole map is on the page, so you can see where every route ends up. Try all three.

Maths · Algebra

Solve it line by line

Pick an equation, then step through it. Each line shows the move you make to both sides, and why.

GoalFind x in two one-step equations

x + 17 = 53

One step to undo: 17 has been added to x. The additive inverse of 17 is −17, because 17 + (−17) = 0.

Step 1 of 6

One step to undo: 17 has been added to x. The additive inverse of 17 is −17, because 17 + (−17) = 0.

Watch out: Whatever you do to one side, you do to the other. If a move only happens on one side, the equation is no longer balanced.

Maths · Like terms

What makes terms “like”?

Before you can solve something like 4x + 5 + 2x + 9 = 38, you have to tidy up the left-hand side by collecting like terms. So what actually makes two terms “like”?

Which of these is closest to what you think right now?

4x and 4y are like terms. They've both got a 4 in front.−2x² and −2x are like terms. They both have an x.4xy and xy can't be like terms. One has a 4 in front and the other has nothing.Like terms have exactly the same variables with the same powers. 4xy and xy match, and so do plain numbers like 3 and 8.

How sure are you?

I'm sureI think soI'm guessing

4x and 4y are like terms. They've both got a 4 in front.

This is a common misconception

It's tempting, but the number in front (the coefficient) isn't what decides it. 4x is made of x and 4y is made of y, and those are different variables, so they can't be collected into a single term.

−2x² and −2x are like terms. They both have an x.

This is a common misconception

Nearly, but the letter has to match with the same power as well. x² and x have different exponents, so −2x² and −2x are not like terms.

4xy and xy can't be like terms. One has a 4 in front and the other has nothing.

This is a common misconception

A term written xy does have a number in front: its coefficient is 1 (just as −x means −1x). Both terms are made from the same variables with the same powers, so 4xy and xy are like terms.

Like terms have exactly the same variables with the same powers. 4xy and xy match, and so do plain numbers like 3 and 8.

This matches the evidence

Yes. The number in front can be anything, but the letters and their powers must match exactly. Constants (terms with no variable) are like terms of each other.

Like terms have the same set of variables with the same exponents, whatever number is in front. That's what lets you tidy 4x + 5 + 2x + 9 into 6x + 14 and get it ready to solve.

Maths · Getting ready to solve

Tidy it, solve it, check it

A regular pentagon sits on top of a rectangle. Going all the way round the whole shape there are five sides of length (x + 1) cm and two sides of length (2x + 3) cm. The perimeter is 83 cm. Find x.

5(x + 1) + 2(2x + 3) = 83Add up the five short sides and the two long sides, and set the total equal to the perimeter. It isn't in the form ax + b = c yet: there are brackets and lots of terms to tidy.
missing step

Which line is step 2?

5x + 5 + 4x + 6 = 835x + 1 + 4x + 3 = 835x + 5 + 2x + 3 = 83

Maths · Choosing your first step

Which first step is easiest?

Every one of these can be solved more than one way. Pick the equation, then pick the first step that makes the working simplest.

Still to sort

Divide by a first (0)

Every term divides exactly by a

Where the line is: Only when ALL the terms share a common factor with a. Otherwise dividing creates fractions.

Add the additive inverse of b first (0)

Dividing would bring in fractions

Where the line is: When a doesn't go into b and c, isolate the x term first and divide last.

Multiply by the LCM first (0)

The equation is full of fractions

Where the line is: Multiply every term by the lowest common multiple of the denominators to clear all the fractions at once.

6 of 6 still to sort.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two jobs stuck to one unknown, and more than one way to free it.

What you need to know

An equation shows two expressions that are equal, so whatever you do to one side you do to the other.
A two-step equation has the form ax + b = c: x is multiplied by a and has b added, and you meet a, b and c as constants.
Make your first step isolate the term with the unknown: add the additive inverse of b, or divide both sides by a.
Equations with brackets or like terms are tidied into ax + b = c first (expand, then collect).
Check your answer by substituting it back into the original equation.

The big picture

To solve ax + b = c, free the x term first by adding the additive inverse of b or dividing by a, then finish the job. Tidy brackets and like terms into that form before you start, and check by putting the answer back in.

Key points

1The additive inverse of a number is the number that adds to it to make zero (the additive inverse of 17 is −17). The reciprocal of a number is its multiplicative inverse (the reciprocal of 7 is 1/7).

2Adding the additive inverse of b first, or dividing by a first, both give the same solution as long as the working is accurate.

3To undo a subtraction you add; to undo a division you multiply.

4A solution can be a fraction or a negative number, and the unknown can be on the right-hand side.

5Dividing first is efficient when every term shares a common factor; with fractions, multiply every term by the lowest common multiple of the denominators.

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Worked example

Problem

Solve 7x + 3 = 59, and check your answer.

⚠ Watch out

Undoing the wrong thing. In 2x − 10 = 24, the − 10 is undone by adding 10, which gives 2x = 34 and x = 17. Adding −10 instead gives 2x − 20 = 14, which is still a two-step equation: you've moved no closer to x.

🧠

Memory hook

Free the x term, then free x. First get the x term standing alone, then undo what's left on it. Do the same to both sides, and check at the end.

✓

Check yourself

Solve 5x + 3 = 38, then prove it by putting your answer back in. (Answer: 5x = 35, so x = 7, and 5 × 7 + 3 = 38.)

Flashcards

(14)

What does an equation show?

Two expressions that are equal to each other.

What is the additive inverse of a number?

The number that, added to it, gives zero. The additive inverse of 17 is −17.

What is the reciprocal of a number?

Its multiplicative inverse. The reciprocal of 7 is 1/7. Any non-zero number has one.

In ax + b = c, what are a, b, c and x?

a, b and c are constants and x is the unknown. a is multiplying x and b is added to it.

What should the first step of a two-step equation aim to do?

Isolate the term with the unknown, by adding the additive inverse of b or by dividing both sides by a.

How do you undo a subtraction, as in 2x − 10 = 24?

Add 10 to both sides. −10 and +10 make a zero pair, so you get 2x = 34.

What is a zero pair?

A number and its additive inverse, like +4 and −4. Together they make zero, so they cancel.

What are like terms?

Terms with the same set of variables and the same exponents, like 4x, 3x and −2x, or 4xy and xy. Plain numbers are like terms of each other.

What is the coefficient of x in the term x? And in −x?

1 and −1. A term written x means 1x, and −x means −1x.

Does it matter whether you divide by a first or add the additive inverse of b first?

No. As long as the working is accurate, both orders reach the same solution.

When is dividing by a first the efficient choice?

When every term shares a common factor with a. Otherwise dividing brings in fractions.

How do you clear the fractions from an equation?

Multiply every term on both sides by the lowest common multiple of the denominators.

What do you do with brackets before collecting like terms?

Expand them first. For example, 2(3x + 4) = 25 becomes 6x + 8 = 25.

How can you check a solution?

Substitute it back into the original equation. If both sides are equal, it is correct.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Which first move would you make?

Solve it line by line

What makes terms “like”?

Tidy it, solve it, check it

Which first step is easiest?

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