03 · Put it into practice

Keep the equation in balance

Solve two-step linear equations—and explain why each step preserves the solution.

28 minutesLearn → explore → practice

BY THE END, YOU WILL BE ABLE TO

Explain equality as two expressions with the same value.

Preserve equality by applying the same operation to both sides.

Solve $ax+b=c$ for $a\ne0$ and verify the result.

Translate a short practical problem into an equation.

Before we begin

You’ll use: Signed arithmetic. Inverse operations and reverse order.

Try this quick warm-up. If it feels unfamiliar, follow the hint or revisit the previous lesson.

Which number, multiplied by 4, gives 20?

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01 / BUILD AN INTUITION

Start with the idea

An equals sign is a claim: the expression on the left has exactly the same value as the expression on the right. It is not an instruction to write an answer. For example, 8 = 5 + 3 is just as valid as 5 + 3 = 8.

Imagine two balanced pans. If both pans have equal mass and you remove the same mass from each, they remain equal. Removing mass from just one pan breaks the balance.

An equation such as 2x + 3 = 11 describes two unknown blocks and three unit weights balancing eleven unit weights. Remove three units from each side. Two blocks now balance eight units. Split both sides into two equal groups: each block is four units.

A LITTLE MORE PRECISELY

An equation states that two expressions are equal. A solution is a value of the variable for which that statement is true. Adding or subtracting the same quantity on both sides preserves the solution set. Multiplying or dividing both sides by the same nonzero constant also preserves it. For $ax+b=c$ with $a\ne0$ , subtract $b$ and divide by $a$ to obtain $x=(c-b)/a$ .

02 / SEE WHAT CHANGES

Make the idea move

THE EQUATION BALANCE2x + 3 = 11

Value of x 4Constant b 3

Find x so the pans balance. Apply the same subtraction to both sides: a balanced equation stays balanced. For b = 3, the solution is x = 4.

03 / THINK IT THROUGH

Three examples, step by step

EXAMPLE 1

Start with one operation

Solve $x+5=12$ . We want x alone.
Subtract 5 from both sides: $x+5-5=12-5$ .
Simplify: $x=7$ . Substitute into the original: $7+5=12$ .

EXAMPLE 2

Two steps, one reason at a time

Solve $3x+4=19$ . The last operation on x is adding 4.
Subtract 4 from both sides: $3x=15$ . Equality is preserved because both values decrease by 4.
Divide both sides by 3: $x=5$ . Check: $3(5)+4=19$ .

EXAMPLE 3

An unfamiliar application

A taxi charges £4 initially and £3 per kilometre. The total is £25. Let x be the distance in kilometres.
The cost model is $3x+4=25$ . Subtract 4 from both sides: $3x=21$ .
Divide both sides by 3: $x=7$ . The journey was 7 km. Check: £4 + 7 × £3 = £25. The units also agree.

04 / LET’S TRY ONE TOGETHER

Your turn, with a little guidance

Solve 4x − 6 = 18. First add 6 to both sides, then find x.

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05 / MAKE IT YOURS

Independent practice

Work on paper if it helps. Give a number, decimal, or fraction such as 3/2. Correct answers to every independent exercise complete this lesson; placement mastery requires a separate assessment.

PRACTICE 1 OF 5

Solve 2x + 7 = 21.

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PRACTICE 2 OF 5

Solve 5x − 8 = 12.

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PRACTICE 3 OF 5

Solve −3x + 2 = 14.

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PRACTICE 4 OF 5

From 3x + 5 = 20, subtract 5 from BOTH sides. What number is on the right?

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PRACTICE 5 OF 5

A membership costs £6 plus £4 per visit. You pay £34. How many visits?

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Easy mistakes. Useful lessons.

“3x + 4 = 19 becomes 3x = 19.”

Removing 4 only on the left destroys equality. Subtract 4 on the right too, giving 3x = 15.

“3x = 15 means x = 12.”

3x means 3 multiplied by x, not 3 added to x. Divide by 3 to undo multiplication.

“A correct final number proves every intermediate step was valid.”

Errors can cancel by coincidence. Explain each operation and substitute into the original equation.

TAKE THESE IDEAS WITH YOU

The knowledge check

Close the examples and explain each point below in your own words. Then solve one practice question again without a hint.

Treat equality as a balance, not a command.
Apply the same operation to both sides. Never divide by zero.
Undo addition or subtraction, then multiplication.
Check your solution in the original context.

If you cannot explain a step yet, return to the visualization and change one value. What stays the same, and why?

YOUR NEXT STEP

See how the ideas fit together

Take an eight-question quiz, or demonstrate all four skills with the placement challenge.

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