1.4 Properties of Algebra (Review)

Terrance Berg

4 1.4 Properties of Algebra (Review)

When doing algebra, it is common not to know the value of the variables. In this case, simplify where possible and leave any unknown variables in the final solution. One way to simplify expressions is to combine like terms.

Like terms are terms whose variables match exactly, exponents included. Examples of like terms would be $3xy$ and $-7xy,$ $3a^2b$ and $8a^2b,$ or −3 and 5. To combine like terms, add (or subtract) the numbers in front of the variables and keep the variables the same.

Example 1.4.1

Simplify $5x - 2y - 8x + 7y.$

$\begin{array}{rl} 5x - 8x \text{ and } -2y + 7y & \text{Combine like terms} \\ \\ -3x + 5y & \text{Solution} \end{array}$

Example 1.4.2

Simplify $8x^2 - 3x + 7 - 2x^2 + 4x - 3.$

$\begin{array}{rl} 8x^2 - 2x^2, -3x + 4x, \text{ and } 7 - 3 & \text{Combine like terms} \\ \\ 6x^2 + x + 4 & \text{Solution} \end{array}$

When combining like terms, subtraction signs must be interpreted as part of the terms they precede. This means that the term following a subtraction sign should be treated like a negative term. The sign always stays with the term.

Another method to simplify is known as distributing. Sometimes, when working with problems, there will be a set of parentheses that makes solving a problem difficult, if not impossible. To get rid of these unwanted parentheses, use the distributive property and multiply the number in front of the parentheses by each term inside.

$\text{Distributive Property: } a(b + c) = ab + ac$

Several examples of using the distributive property are given below.

Example 1.4.3

Simplify $4(2x-7).$

$\begin{array}{rl} 4(2x-7)& \text{Multiply each term by } 4. \\ \\ 8x-28 & \text{Solution} \end{array}$

Example 1.4.4

Simplify $-7(5x-6).$

$\begin{array}{rl} -7(5x-6) & \text{Multiply each term by }-7. \\ \\ -35x+42 & \text{Solution} \end{array}$

In the previous example, it is necessary to again use the fact that the sign goes with the number. This means −6 is treated as a negative number, which gives (−7)(−6) = 42, a positive number. The most common error in distributing is a sign error. Be very careful with signs! It is possible to distribute just a negative throughout parentheses. If there is a negative sign in front of parentheses, think of it like a −1 in front and distribute it throughout.

Example 1.4.5

Simplify $-(4x-5y+6).$

$\begin{array}{rl} -(4x-5y+6) & \text{Negative can be thought of as }-1. \\ \\ -1(4x-5y+6) & \text{Multiply each term by }-1. \\ \\ -4x+5y-6 & \text{Solution} \end{array}$

Distributing throughout parentheses and combining like terms can be combined into one problem. Order of operations says to multiply (distribute) first, then add or subtract (combine like terms). Thus, do each problem in two steps: distribute, then combine.

Example 1.4.6

Simplify $3x-2(4x-5).$

$\begin{array}{rl} 3x-2(4x-5) & \text{Distribute }-2, \text{ multiplying each term.} \\ \\ 3x-8x+10 & \text{Combine like terms }3x-8x. \\ \\ -5x+10 & \text{Solution} \end{array}$

Example 1.4.7

Simplify $5+3(2x-4).$

$\begin{array}{rl} 5+3(2x-4) & \text{Distribute 3, multiplying each term.} \\ \\ 5+6x-12 & \text{Combine like terms }5-12. \\ \\ -7+6x & \text{Solution} \end{array}$

In Example 1.4.6, −2 is distributed, not just 2. This is because a number being subtracted must always be treated like it has a negative sign attached to it. This makes a big difference, for in that example, when the −5 inside the parentheses is multiplied by −2, the result is a positive number. More involved examples of distributing and combining like terms follow.

Example 1.4.8

Simplify $2(5x-8)-6(4x+3).$

$\begin{array}{rl} 2(5x-8)-6(4x+3) & \text{Distribute 2 into the first set of parentheses and }-6\text{ into the second.} \\ \\ 10x-16-24x-18 & \text{Combine like terms }10x-24x\text{ and }-16-18. \\ \\ -14x-34 & \text{Solution} \end{array}$

Example 1.4.9

Simplify $4(3x-8)-(2x-7).$

$\begin{array}{rl} 4(3x-8)-(2x-7) & \text{The negative sign in the middle can be thought of as }-1. \\ \\ 4(3x-8)-(2x-7) & \text{Distribute 4 into the first set of parentheses and }-1\text{ into the second.} \\ \\ 12x-32-2x+7 & \text{Combine like terms }12x-2x\text{ and }-32+7. \\ \\ 10x-25& \text{Solution} \end{array}$