Simplifying Algebraic Expressions

Final answer:

$5x − 2$

Always write your answer in its simplest form.

Check your solution (important habit)

Pick a value for $x$ to check both expressions give the same result.

Let $x = 1$

Original expression:

$3(1) + 5 + 2(1) − 7 = 3 + 5 + 2 − 7 = 3$

Simplified expression:

$5(1) − 2 = 5 − 2 = 3$

Both expressions give the same result for x = 1. Substitution is a useful check for mistakes, but matching at one value does not prove that two expressions are equal for every value of x. Here, collecting like terms using the rules of algebra establishes the simplification.

Products, powers and algebraic reasoning

Worked example — More than one kind of term

Simplify 4a² + 3ab − a² + 2ab − 5.

Group matching letter-and-power parts: (4 − 1)a² + (3 + 2)ab − 5.

3a² + 5ab − 5. The a² and ab terms cannot be combined.

Worked example — Multiply and divide terms

In 3a × 4ab, multiply the numbers and then the letters.

3 × 4 × a × a × b = 12a²b.

For 18x³ ÷ 6x, divide the coefficients and cancel one factor x: 18 ÷ 6 = 3 and x³ ÷ x = x².

The result is 3x², provided x ≠ 0. The symbol ≠ means “not equal to”; the original divisor cannot be zero.

Addition works differently: 3a + 4a = 7a, not 7a².

Worked example — Apply index laws

A positive whole-number index tells us how many equal factors are multiplied. x² × x³ has five factors x, so it is x⁵.

x⁵ ÷ x² = x³ for x ≠ 0, because two factors cancel.

(2x²)³ = (2x²)(2x²)(2x²) = 8x⁶. Cube the coefficient as well as the power of x.

For a non-zero base, x⁰ = 1: x³ ÷ x³ is both 1 and x⁰.

Negative integer powers — Foundation & Higher

For x ≠ 0, x⁻² means 1/x². A negative index does not mean a negative answer.

x³ × x⁻⁵ = x⁻² = 1/x².

Higher extension — Fractional powers

For positive x, x^1/2 means √x. Thus x^1/2 × x = x^3/2: add ½ and 1.

Use the same-base rules only when the bases match. They do not turn x² × y³ into a single power of x. For the numerical index laws and root methods, see Powers and Roots.

Worked example — Show two expressions are equivalent

Show that 5x + 4 − 2x and 3(x + 1) + 1 have equal values for every x.

Collect terms in the first: 5x − 2x + 4 = 3x + 4.

Expand the second: 3x + 3 + 1 = 3x + 4.

Both simplify to the same expression. This establishes an identity; checking just one number would not.

Higher extension — A short algebraic proof

Prove that the sum of three consecutive integers is a multiple of 3. Let n be any integer. The next two integers are n + 1 and n + 2.

n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1).

Since n + 1 is an integer, this is 3 times an integer. That proves the claim for every starting integer.

A few numerical examples can suggest a pattern, but they cannot prove a statement about all integers. Continue to Expanding Brackets for the main algebraic-proof method.

More practice — expressions and reasoning

1. Simplify 6x² − 2x + x² + 5x − 4.

2. Simplify 5a × 2ab and 20y⁴ ÷ 5y².

3. Simplify (3x²)² and x³ × x⁴.

4. Show that 2(3x + 1) − x is equivalent to 5x + 2.

5. Simplify x² ÷ x⁵ for x ≠ 0, using a positive index in your answer.

6. Higher: prove that the sum of two consecutive odd integers is divisible by 4.

Frequently Asked Questions

What are like terms?

Like terms have exactly the same variables raised to the same powers.

For example:

$3x$ and $−5x$ are like terms

$2a²$ and $7a²$ are like terms

$x$ and $x²$ are NOT like terms

Why can we add $3x$ and $5x$ but not $3x$ and $5$ ?

Because $3x$ and $5x$ represent quantities of $x$, while $5$ is just a number with no variable.