2.02 Simplify expressions and distributive property

So far we have used algebra tiles to generate equivalent expressions and the order of operations and the properties of real numbers to simplify expressions. Simplifying an expression involves rewriting an algebraic expression in its most basic form.

To solve problems with mixed operations, we need apply the order of opertions and the properties of real numbers.

Recall the order of operations:

1. Evaluate operations inside grouping symbols such as parentheses.

2. Evaluate exponents.

3. Evaluate multiplication or division, from left to right.

4. Evaluate addition or subtraction, from left to right.

Recall the properties of real numbers:

Examples

Remember that the order of operations calls for us to evaluate expressions inside grouping symbols (such as parentheses) first. However, sometimes the terms inside the grouping symbol cannot be combined. Is there a different way that we might rewrite the expression?

The expression 2\left(4x+1\right) is the same as saying 2 groups of 4x+1, or 2 groups of 4x and 2 groups of 1.

We can represent these different groupings visually with algebra tiles:

2 groups of 4x+1

2 groups of 4x and 2 groups of 1

We can also rewrite this expression using the associative and commutative properties of addition.

Another way to demonstrate this is with the area of rectangles. The area of a rectangle can be found by multiplying its length by its width.

The dimensions of this rectangle are 2 and 4x+1, so its area is 2\left(4x+1\right).

We can find the total area of the large rectangle by adding the area of each of the smaller rectangles.

This clearly shows the distributive property.

For example,

The 5 is multiplied by each term inside the parentheses. To distribute the 5, you multiply it by each term separated by the plus sign. In this case, the x and the 6 are each multiplied by the 5.

Examples

Example 2

Consider the expression 4\left(t+6\right). Use a model to simplify this expression.

Worked Solution

Create a strategy

We can use an area model to represent this expression. The dimensions of this rectangle are 4 and \left(t+6\right) so its area is 4\left(t+6\right).

Apply the idea

The area model will look like this:

We can find the total area of the large rectangle by adding the area of each of the smaller rectangles.

\displaystyle \text{Area Large Rectangle}	\displaystyle =	\displaystyle \text{Area Small Rectangle}_1 + \text{Area Small Rectangle}_2
\displaystyle 4(t+6)	\displaystyle =	\displaystyle 4 \cdot t + 4 \cdot 6
	\displaystyle =	\displaystyle 4t+24

Reflect and check

We can also represent the expression using algebra tiles. The expression 4\left(t+4\right) can be represented as 4 groups of \left(t+6\right).

or 4 groups of t and 4 groups of 6.

Watch question walkthrough

Example 3

Simplify the expressions using the distributive property.

a

-2\left(3x-1\right)

Worked Solution

Create a strategy

Use the distributive property a\left(b+c\right)=a\cdot b+a \cdot c.

Apply the idea

\displaystyle -2\left(3x-1\right)	\displaystyle =	\displaystyle \left(-2\right) \cdot 3x+ \left(-2\right) \cdot \left(-1\right)	Distributive property
	\displaystyle =	\displaystyle -6x+2	Evaluate the multiplication

Watch question walkthrough

b

-\left(5-s\right)

Worked Solution

Create a strategy

Use the distributive property a\left(b+c\right)=a\cdot b+a\cdot c. The negative sign outside the parentheses represents multiplying by -1.

Apply the idea

\displaystyle -\left(5-s\right)	\displaystyle =	\displaystyle -1\left(5-s\right)	Rewrite using -1
	\displaystyle =	\displaystyle -1 \cdot 5 -\left(-1\right)\cdot s	Distributive property
	\displaystyle =	\displaystyle -5-\left(-s\right)	Evaluate the multiplication
	\displaystyle =	\displaystyle -5+s	Simplify the adjacent signs

Watch question walkthrough

c

- 6.1 \left(3.5y + 4.5\right)

Worked Solution

Create a strategy

Use the distributive property.

Apply the idea

\displaystyle - 6.1 \left(3.5y + 4.5\right)	\displaystyle =	\displaystyle \left(-6.1\right) \cdot 3.5y + \left(-6.1\right) \cdot 4.5	Distributive property
	\displaystyle =	\displaystyle -21.35y + \left(-27.45 \right)	Evaluate the multiplication
	\displaystyle =	\displaystyle -21.35y - 27.45	Inverse property of addition

Reflect and check

We can check our work by substituting a value for y and confirming that the original expression and the simplified expression yield the same result.

Let's let y=1 and see if both expressions evaluate to the same number.

\displaystyle -6.1\left(3.5\left(1\right)+4.5\right)	\displaystyle =	\displaystyle -21.35\left(1\right)-27.45
\displaystyle -6.1\left(3.5+4.5\right)	\displaystyle =	\displaystyle -21.35-27.45
\displaystyle -6.1\left(8\right)	\displaystyle =	\displaystyle -48.8
\displaystyle -48.8	\displaystyle =	\displaystyle -48.8

Since both expressions evaluate to -48.8 when y=1, we can be confident that these two expressions are equivalent.

Watch question walkthrough

d

\dfrac{1}{4}\left(x-4\right)-\dfrac{1}{5}x

Worked Solution

Create a strategy

Use the distributive property to rewrite the expression in parentheses. Then combine like terms and make sure all fractions are fully simplified.

Apply the idea

\displaystyle \frac{1}{4}\left(x-4\right)-\frac{1}{5}x	\displaystyle =	\displaystyle \frac{1}{4}x-\frac{1}{4}\cdot 4-\frac{1}{5}x	Distributive property
	\displaystyle =	\displaystyle \frac{1}{4}x-1-\frac{1}{5}x	Multiplicative inverse
	\displaystyle =	\displaystyle \frac{1}{4}x-\frac{1}{5}x-1	Commutative property of addition
	\displaystyle =	\displaystyle \left( \frac{1}{4}\cdot \frac{5}{5} \right) x - \left( \frac{1}{5} \cdot \frac{4}{4}\right)x -1	Multiplicative identity
	\displaystyle =	\displaystyle \frac{5}{20}x-\frac{4}{20}x-1	Evaluate the multiplication
	\displaystyle =	\displaystyle \frac{1}{20} x-1	Combine like terms

Reflect and check

\dfrac{1}{20}x-1 can also be written as \dfrac{x}{20}-1. Always check if fractions are fully simplified.

Watch question walkthrough

Example 4

A student incorrectly used the distributive property and wrote 7\left(4x+3\right)=28x+3.

Which one of the following is the best explanation to help the student correct their error?

A

They have multiplied 4x and 7 rather than adding them.

B

They have forgotten to multiply the second part of the sum, 3, by the number outside the brackets, 7.

C

They have added 4x and 7 rather than multiplying them.

D

They have multiplied the wrong term in the sum by 7. They should multiply 3, instead of 4x, by 7.

Worked Solution

Create a strategy

Use the distributive property a\left(b+c\right)=a\cdot b+a \cdot c.

Apply the idea

The correct option is B, because they have forgotten to multiply the second part of the sum, 7, by the number outside the brackets, 3.

Watch question walkthrough