Expressions and equivalence

An expression names a quantity

Three identical notebooks cost x dollars each. Their total cost is 3x dollars. Adding a delivery fee of 2 dollars gives the expression 3x + 2.

An expression has a value when its variables have values. An equation states that two expressions are equal. Thus 3x + 2 is an expression, while 3x + 2 = 17 is an equation that may be true or false for a particular x.

For x = 5, the expression has value 3(5) + 2 = 17, so the equation is true. For x = 4, its value is 14, so that equation is false. An equation’s solution is a value that makes its equality true.

Equivalent expressions agree for every allowed input

The expressions 2(x + 3) and 2x + 6 are equivalent: multiplying a sum by two multiplies both parts by two. Draw two bags, each holding x counters and three loose counters. Together they hold 2x + 6 counters.

This equality is an identity: it holds for every real x. By contrast, 2(x + 3) = 2x + 3 is false. It counts the three loose counters in only one bag.

Checking one input can disprove an identity. It cannot normally prove one. At x = 0, x² = x is true, but at x = 2 it is false. A valid distributive argument explains why the correct identity works for every input.

Check your understanding 1 The whole bracket is multiplied

Expand -3(2x-5), then check your result at x = 1.

Expanded expression and substitution check

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-3(2x-5) = -6x + 15. The second product is (-3)(-5) = +15. At x = 1, the original expression is -3(-3) = 9 and the expanded expression is -6 + 15 = 9. The distributive law proves the identity; the substitution catches a possible sign slip.

If you wrote -6x-15, compare the two values at x = 1: 9 and -21. Then try -2(x-4) yourself: -2x + 8.

Collect terms that count the same thing

3x + 2x = 5x because three lots of x and two lots of x are five lots of x. But 3x + 2 cannot be combined into 5x: the 2 is not two lots of x.

Keep signs attached to terms:

The regrouping uses addition of signed quantities. It does not move a term across an equality sign.

Factorising reverses expansion. In 6x + 9, both terms contain a factor of three, so 6x + 9 = 3(2x + 3). Expanding the result checks that nothing was lost.

Choose a form for a purpose

An equivalent value and a requested form are separate matters. 2(x + 3) and 2x + 6 agree, but only 2x + 6 is expanded. If asked to factorise 2x + 6 fully, the useful answer is 2(x + 3).

A checker may verify equality without verifying an explanation. Explain on paper or in the working field when a question asks you to justify a step. An unsupported argument remains unverified; it is not made correct by the final expression.

Check your understanding 2 A changed task

Simplify 5(2y-1)-3(y + 2). Then decide whether the result gives a single value of y.

Simplification and whether y is determined

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10y-5-3y-6 = 7y-11. This is an expression. No equality or other condition has been given, so there is no single value of y to solve for. If the question instead said 7y-11 = 10, then y = 3 would satisfy it.