3.04 Simplifying expressions

Lesson

We simplify algebraic expressions following the same order of operations as for numeric expressions. That is,

1. Simplify any expression in brackets first.
2. Simplify any powers.
3. Simplify any multiplication or division from left to right.
4. Simplify any addition or subtraction from left to right.

Worked example

example 1

Simplify $\frac{8xy+16xy}{30yz-9yz}$8xy+16xy30yz9yz.

Think: The numerator and denominator count as being inside brackets, so we want to simplify each of them before we simplify the fraction.

Do:

 $\frac{8xy+16xy}{30yz-9yz}$8xy+16xy30yz−9yz​ $=$= $\frac{24xy}{30yz-9yz}$24xy30yz−9yz​ Simplify the addition in the numerator $=$= $\frac{24xy}{21yz}$24xy21yz​ Simplify the subtraction in the denominator $=$= $\frac{8x}{7z}$8x7z​ Simplify the fraction by cancelling the common factor of $3y$3y

Reflect: The order of operations was the same as for a numerical expression.

Summary

We simplify algebraic expressions following the same order of operations as for numeric expressions. That is,

1. Simplify any expression in brackets first.
2. Simplify any powers.
3. Simplify any multiplication or division from left to right.
4. Simplify any addition or subtraction from left to right.

Practice questions

Question 1

Simplify the expression $2t+4\times8t$2t+4×8t.

Question 2

Simplify the expression $12u\div3-2u$12u÷​32u.

Question 3

Simplify the expression $\left(13a+15a\right)\div7$(13a+15a)÷​7.

Equivalent expressions

Simplified expressions are equivalent to the original expression. But there are many ways to write an equivalent expression.

Worked example

example 2

Write three equivalent expressions to the expression: $4(2x-6)+8x.$4(2x6)+8x.

Think: One way we could find an equivalent expression is just by separating the $8x$8x into two terms.

Do: So an equivalent expression would be: $4(2x-6)+3x+5x$4(2x6)+3x+5x.

Think: Another way we could find an equivalent expression is just by expanding the brackets.

Do

 $4(2x-6)+8x$4(2x−6)+8x $=$= $8x-24+8x$8x−24+8x

Think: To get a third equivalent expression we can just simplify the previous expression.

Do: $8x-24+8x=16x-24$8x24+8x=16x24.

So an equivalent expression does not have to be in simplest form.

Outcomes

9.C1.3

Compare algebraic expressions using concrete, numerical, graphical, and algebraic methods to identify those that are equivalent, and justify their choices.

9.C1.4

Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts.