1.11 Simplifying algebraic expressions

Identifying equal expressions

We've already learned about the components of an expression, how to identify like terms and how to identify missing terms that make number sentences equivalent or equal. Now we are going to build on this knowledge and learn how to generate our own equivalent expressions with number sentences and word problems. 

It's really important to remember the rules of algebra because there are many ways of writing equivalent algebraic expressions. For example, $2a+7$2a+7 could be written as $7+2a$7+2a, $a\times2+7$a×2+7 or $7+a\times2$7+a×2 just to name a few. 

It's good to be flexible so that no matter how an algebraic expression is written, you'll know what it means. Let's look through some examples and remember, there is more than one way to write an equivalent algebraic expression.

 

Practice questions

Question 1

Question 2

 

Simplifying algebraic expressions - basic operations

We've already learned about the concept of negative numbers and how to simplify and evaluate expressions with integers.

Now we are going to look at working with algebraic expressions that have positive and negative terms.

There are a few important things to remember when working with integers:

	Example
Multiplying a positive term and a negative term gives a negative answer	$\left(-2f\right)\times5g=-10fg$(−2f)×5g=−10fg
Multiplying two negative terms gives a positive answer	$-6\times\left(-7c\right)=42c$−6×(−7c)=42c
Subtracting a negative term is the same as adding a positive term	$3-\left(-2a\right)$3−(−2a) is the same as $3+2a$3+2a
Adding a negative term is the same as subtracting the term	$h+\left(-3\right)$h+(−3) is the same as $h-3$h−3

 

 

Practice questions

Question 3

Question 4

 

Simplify expressions involving the distributive property

We have learned how to:

• distribute parentheses using the distributive law,
• collect like terms, and
• perform operations with positive and negative algebraic terms.

Now we can use all these skills to simplify expressions which involve the four operations and the distributive property. 

When there is an expression involving a mixture of $+$+, $-$−, $\times$×, $\div$÷ and distributing terms in parentheses, we need to follow the correct order of operations.

1) If we see any parentheses, we need to distribute them first.

For example, in the expression $5\left(x+3\right)-2x$5(x+3)−2x, we need to distribute $5\left(x+3\right)$5(x+3) before we consider the other term. We get $5x+15-2x$5x+15−2x.

2) Collect the like terms to simplify.

After distributing the parentheses included in the expression $5\left(x+3\right)-2x$5(x+3)−2x, we got $5x+15-2x$5x+15−2x. Since there are now some like terms, we can combine these and simplify the expression even more. Doing this we get $3x+15$3x+15.

 

More than one set of parentheses

If there is more than one set of parentheses, we can distribute them all in the one step.

Here is an expression which involves distributing the terms inside more than one set of parentheses:

$4\left(2a-3b\right)+a\left(2+b\right)$4(2a−3b)+a(2+b)

We can distribute both sets of parentheses at the same time. We get:

$8a-12b+2a+ab$8a−12b+2a+ab

From here, we can collect the like terms $8a$8a and $2a$2a. Doing this, we get:

$10a-12b+ab$10a−12b+ab

 

The sign is important

When distributing the parentheses in an expression, we need to be careful to multiply negative terms correctly.

From our work with positive and negative numbers, we know that:

• multiplying two negatives gives us a positive result
• multiplying a positive and a negative gives us a negative result

In the expression $8y-5\left(y-3\right)-2$8y−5(y−3)−2, we need to distribute the terms inside the parentheses first.

We get $8y-5y+15-2$8y−5y+15−2.

Notice the positive $15$15.

We can then combine like terms to get $3y+13$3y+13.

 

 

Practice questions

Question 5

Question 6

Question 7