Section 2.9 Simplifying Expressions
We know that if we have two apples and add three more, then our result is the same as if we'd had three apples and added two more. In this section, we'll formally define and extend these basic properties we know about numbers to variable expressions.
Subsection 2.9.1 Identities and Inverses
We will start with some definitions. The number \(0\) is called the additive identity. If the sum of two numbers is the additive identity, \(0\text{,}\) these two numbers are called additive inverses. For example, \(2\) is the additive inverse of \(-2\text{,}\) and the additive inverse of \(-2\) is \(2\text{.}\)
Similarly, the number \(1\) is called the multiplicative identity. If the product of two numbers is the multiplicative identity, \(1\text{,}\) these two numbers are called multiplicative inverses. For example, \(2\) is the multiplicative inverse of \(\frac{1}{2}\text{,}\) and the multiplicative inverse of \(-\frac{2}{3}\) is \(-\frac{3}{2}\text{.}\) The multiplicative inverse is also called reciprocal.
Subsection 2.9.2 Introduction to Algebraic Properties
Commutative Property.
When we compute the area of a rectangle, we generally multiply the length by the width. Does the result change if we multiply the width by the length?
We can see \(3\cdot2=2\cdot3\text{.}\) If we denote the length of a rectangle with \(\ell\) and the width with \(w\text{,}\) this implies \(\ell w=w\ell\text{.}\) This is referred to as the commutative property of multiplication. The commutative property also applies to addition, as in \(1+2=2+1\text{,}\) where it is called the commutative property of addition. However, there is no commutative property of subtraction or division, as \(2-1\ne1-2\text{,}\) and \(\frac{4}{2}\ne\frac{2}{4}\text{.}\)
Associative Property.
Let's extend that example to a rectangular prism with width \(w=4\,\text{cm}\text{,}\) depth \(d=3\,\text{cm}\text{,}\) and height \(h=2\,\text{cm}\text{.}\) To compute the volume of this solid, we multiply the width, depth and height, which we write as \(wdh\text{.}\)
In the following figure, on the left side, we multiply the length and width first, and then multiply the height; on the right side, we multiply the width and height first, and then multiply the length. Let's compare the products.
We can see \((wd)h=w(dh)\text{.}\) This is known as the associative property of multiplication. The associative property also applies to addition, as in \((1+2)+3=1+(2+3)\text{,}\) which is called the associative property of addition. However, there is no associative property of subtraction, as \((3-2)-1\ne3-(2-1)\text{.}\)
Distributive Property.
The final property we'll explore is called the distributive property, which involves both multiplication and addition. To conceptualize this property, let's consider what happens if we buy 3 boxes that each contain one apple and one pear. This will have the same total cost as if we'd bought 3 apples and 3 pears. We write this algebraically:
Visually, we can see that it's just a means of re-grouping: \(3(\apple+ \pear) = 3(\apple)+ 3(\pear)\text{.}\)
Subsection 2.9.3 Summary of Algebraic Properties
Let \(a\text{,}\) \(b\text{,}\) and \(c\) represent real numbers, variables, or algebraic expressions. Then the following properties hold:
- Commutative Property of Multiplication
\(\displaystyle a\cdot b=b\cdot a\)
- Associative Property of Multiplication
\(\displaystyle a\cdot(b\cdot c)=(a\cdot b)\cdot c\)
- Commutative Property of Addition
\(\displaystyle a+b=b+a\)
- Associative Property of Addition
\(\displaystyle a+(b+c)=(a+b)+c\)
- Distributive Property
\(\displaystyle a(b+c)=ab+ac\)
Let's practice these properties in the following exercises.
Checkpoint 2.9.5.
Subsection 2.9.4 Applying the Commutative, Associative, and Distributive Properties
Like Terms.
One of the main ways that we will use the commutative, associative, and distributive properties is to simplify expressions. In order to do this, we need to recognize like terms, as discussed in Section 2.3. We combine like terms when we take an expression lilke \(2a+3a\) and write the result as \(5a\text{.}\) The formal process actually involves using the distributive property:
In practice, however, it's more helpful to think of this as having \(2\) of an object and then an additional \(3\) of that same object. In total, we then have \(5\) of that object.
Example 2.9.6.
Where possible, simplify the following expressions by combining like terms.
\(\displaystyle 6c+12c-5c\)
\(\displaystyle -5q^2-3q^2\)
\(\displaystyle x-5y+4x\)
\(\displaystyle 2x-3y+4z\)
-
All three terms are like terms, so they may combined. We combine them two at a time:
\begin{align*} 6c+12c-5c\amp=18c-5c\\ \amp=13c \end{align*} -
The two terms \(-5q^2\) and \(-3q^2\) are like terms, so we may combine them:
\begin{align*} -5q^2-3q^2 \amp=-8q^2 \end{align*} -
The two terms \(x\) and \(4x\) are like terms, while the other term is different. Using the associative and commutative properties of addition in the first step allows us to place the two like terms next to each other, and then combine them:
\begin{align*} x-5y+4x\amp=x+4x+(-5y)\\ \amp=1x+4x+(-5y)\\ \amp=5x-5y \end{align*}Note the expression \(x\) is the same as \(1x\text{.}\) Usually we don't write the “\(1\)” as it is implied. However, it's helpful when combining like terms to remember that \(x=1x\text{.}\) (Similarly, \(-x\) is equal to \(-1x\text{,}\) which can be helpful when combining \(-x\) with like terms.)
The expression \(2x-3y+4z\) cannot be simplified as there are no like terms.
Adding Expressions.
When we add an expression like \(4x-5\) to an expression like \(3x-7\text{,}\) we write them as follows:
In order to remove the given sets of parentheses and apply the commutative property of addition, we will rewrite the subtraction operation as “adding the opposite” :
At this point we can apply the commutative property of addition and then combine like terms. Here's how the entire problem will look:
Remark 2.9.7.
Once we become more comfortable simplifying such expressions, we will simply write this kind of simplification in one step:
Example 2.9.8.
Use the associative, commutative, and distributive properties to simplify the following expressions as much as possible.
\(\displaystyle (2x+3)+(4x+5)\)
\(\displaystyle (-5x+3)+(4x-7)\)
-
We will remove parentheses, and then combine like terms:
\begin{align*} (2x+3)+(4x+5)\amp=2x+3+4x+5\\ \amp=2x+4x+3+5\\ \amp=6x+8 \end{align*} -
We will remove parentheses, and then combine like terms:
\begin{align*} (-5x+3)+(4x-7)\amp=-5x+3+4x+(-7)\\ \amp=-x+(-4)\\ \amp=-x-4 \end{align*}
Applying the Distributive Property with a Negative Constant.
Applying the distributive property in an expression such as \(2(3x+4)\) is fairly straightforward, in that this becomes \(2(3x)+2(4)\text{,}\) which we then simplify to \(6x+8\text{.}\) Applying the distributive property is a little trickier when subtraction or a negative constant is involved, for example, with the expression \(2(3x-4)\text{.}\) Recalling that subtraction is defined as “adding the opposite,” we can change the subtraction of positive \(4\) to the addition of negative \(4\text{:}\)
Now when we distribute, we obtain:
As a final step, we see that this simplifies to:
Remark 2.9.9.
We can also extend the distributive property to use subtraction, and state that \(a(b-c)=ab-ac\text{.}\) With this property, we would simplify \(2(3x-4)\) more efficiently:
In general, we will use this approach.
Example 2.9.10.
Apply the distributive property to each expression and simplify it as much as possible.
\(\displaystyle -3(5x+7)\)
\(\displaystyle 2(-4x-1)\)
-
We will distribute \(-3\) to the \(5x\) and \(7\text{:}\)
\begin{align*} -3(5x+7)\amp=-3(5x)+(-3)(7)\\ \amp=-15x-21 \end{align*} -
We will distribute \(2\) to the \(-4x\) and \(-1\text{:}\)
\begin{align*} 2(-4x-1)\amp=2(-4x)-2(1)\\ \amp=-8x-2 \end{align*}
Checkpoint 2.9.11.
Subtracting Expressions.
To subtract one expression from another expression, such as \((5x+9)-(3x+2)\text{,}\) we will again rely on the fact that subtraction is defined as “adding the opposite.” To add the opposite of an expression, we will technically distribute a constant factor of \(-1\) and simplify from there:
Remark 2.9.12.
The above example demonstrates how we apply the distributive property in order to subtract two expressions. But in practice, it can be pretty cumbersome. A shorter (and often clearer) approach is to instead subtract every term in the expression we are subtracting, which is shown like this:
In general, we'll use this approach.
Example 2.9.13.
Use the associative, commutative, and distributive properties to simplify the following expressions as much as possible.
\(\displaystyle (-6x+4)-(3x-7)\)
\(\displaystyle (-2x-5)-(-4x-6)\)
-
We will remove parentheses using the distributive property, and then combine like terms:
\begin{align*} (-6x+4)-(3x-7)\amp=-6x+4-3x-(-7)\\ \amp=-6x+4-3x+7\\ \amp=-9x+11 \end{align*} -
We will remove parentheses using the distributive property, and then combine like terms:
\begin{align*} (-2x-5)-(-4x-6)\amp=-2x-5-(-4x)-(-6)\\ \amp=-2x-5+4x+6\\ \amp=2x+1 \end{align*}
Subsection 2.9.5 The Role of the Order of Operations in Applying the Commutative, Associative, and Distributive Properties
When simplifying an expression such as \(3+4(5x+7)\text{,}\) we need to respect the order of operations. Since the terms inside the parentheses are not like terms, there is nothing to simplify there. The next highest priority operation is multiplying the \(4\) by \((5x+7)\text{.}\) This must be done before anything happens with the adding of that \(3\text{.}\) We cannot say \(3+4(5x+7)=\highlight{7}(5x+7)\text{,}\) because that would mean we treated the addition as having higher priority than the multiplication.
So to simplify \(3+4(5x+7)\text{,}\) we will first examine the multiplication of \(4\) with \((5x+7)\text{,}\) and here we may apply the distributive property. After that, we will use the commutative and associative properties:
Example 2.9.14.
Simplify the following expressions using the commutative, associative, and distributive properties.
\(\displaystyle 4-(3x-9)\)
\(\displaystyle 5x+9(-2x+3)\)
\(\displaystyle 5(x-9)+4(x+4)\)
-
We will remove parentheses using the distributive property, and then combine like terms:
\begin{align*} 4-(3x-9)\amp=4-3x-(-9)\\ \amp=4-3x+9\\ \amp=-3x+13 \end{align*} -
We will remove parentheses using the distributive property, and then combine like terms:
\begin{align*} 5x+9(-2x+3)\amp=5x+9(-2x)+9(3)\\ \amp=5x-18x+27\\ \amp=-13x+27 \end{align*} -
We will remove parentheses using the distributive property, and then combine like terms:
\begin{align*} 5(x-9)+4(x+4)\amp=5x-45+4x+16\\ \amp=9x-29 \end{align*}
Checkpoint 2.9.15.
Exercises 2.9.6 Exercises
Review and Warmup
1.
Count the number of terms in each expression.
\({-2x^{2}+6s^{2}+4t^{2}+y}\)
\({-9y^{2}+2t}\)
\({s+8}\)
\({4s+7y+8x}\)
2.
Count the number of terms in each expression.
\({-2x-2x-2t+6x}\)
\({-5y^{2}-8x+6z}\)
\({-9x+7y^{2}+8z}\)
\({-7z}\)
3.
List the terms in each expression.
\({1.6x^{2}}\)
\({4t-0.3x-1.1z}\)
\({y+7.6t}\)
\({-3.1x-7.1y^{2}+2.1x+5.8z^{2}}\)
4.
List the terms in each expression.
\({3.3x}\)
\({-5.9x^{2}-2.3t}\)
\({-2.4z-3s^{2}+1.3+8.9y}\)
\({-5.1s^{2}-6z+1.5x}\)
5.
List the terms in each expression.
\({4.9x+2.3t^{2}}\)
\({-4.3y-3.6z^{2}-7.1z-4.9s}\)
\({4.4y}\)
\({1.4y-2.5z^{2}+2.4t}\)
6.
List the terms in each expression.
\({-8.9x^{2}+8}\)
\({-0.7z+3.1-8.6s}\)
\({6.5s+2.8s+3.3s}\)
\({2.6z}\)
7.
Simplify each expression, if possible, by combining like terms.
\({8x+x-9}\)
\({-9t+z^{2}}\)
\({6z^{2}-t-2t+4t^{2}}\)
\({-2s^{2}+9s^{2}-8y^{2}}\)
8.
Simplify each expression, if possible, by combining like terms.
\({7x^{2}+8s^{2}-8y}\)
\({-x-6y^{2}+4-8y^{2}}\)
\({-3t^{2}-6t^{2}}\)
\({6t^{2}-9t^{2}+9z^{2}}\)
These exercises involve the concepts of like terms and the commutative, associative, and distributive properties.
9.
What is the additive inverse of \(4\text{?}\)
10.
What is the additive inverse of \(6\text{?}\)
11.
What is the multiplicative inverse of \(8\text{?}\)
12.
What is the multiplicative inverse of \(-10\text{?}\)
13.
Use the associative property of addition to write an equivalent expression to \({y+\left(37+q\right)}\text{.}\)
14.
Use the associative property of addition to write an equivalent expression to \({t+\left(80+m\right)}\text{.}\)
15.
Use the associative property of addition to write an equivalent expression to \({7+\left(4+a\right)}\text{.}\)
16.
Use the associative property of addition to write an equivalent expression to \({20+\left(13+c\right)}\text{.}\)
17.
Use the associative property of multiplication to write an equivalent expression to \({10\!\left(9x\right)}\text{.}\)
18.
Use the associative property of multiplication to write an equivalent expression to \({6\!\left(8t\right)}\text{.}\)
19.
Use the commutative property of addition to write an equivalent expression to \({m+92}\text{.}\)
20.
Use the commutative property of addition to write an equivalent expression to \({p+57}\text{.}\)
21.
Use the commutative property of addition to write an equivalent expression to \({8q+22}\text{.}\)
22.
Use the commutative property of addition to write an equivalent expression to \({3y+87}\text{.}\)
23.
Use the commutative property of addition to write an equivalent expression to \({6\!\left(t+52\right)}\text{.}\)
24.
Use the commutative property of addition to write an equivalent expression to \({10\!\left(a+18\right)}\text{.}\)
25.
Use the commutative property of multiplication to write an equivalent expression to \({83c}\text{.}\)
26.
Use the commutative property of multiplication to write an equivalent expression to \({79c}\text{.}\)
27.
Use the commutative property of multiplication to write an equivalent expression to \({20+3y}\text{.}\)
28.
Use the commutative property of multiplication to write an equivalent expression to \({79+7m}\text{.}\)
29.
Use the commutative property of multiplication to write an equivalent expression to \({2\!\left(p+44\right)}\text{.}\)
30.
Use the commutative property of multiplication to write an equivalent expression to \({6\!\left(q+9\right)}\text{.}\)
31.
Use the distributive property to write an equivalent expression to \({8\!\left(y+9\right)}\) that has no grouping symbols.
32.
Use the distributive property to write an equivalent expression to \({5\!\left(t+4\right)}\) that has no grouping symbols.
33.
Use the distributive property to write an equivalent expression to \({-10\!\left(a+4\right)}\) that has no grouping symbols.
34.
Use the distributive property to write an equivalent expression to \({-4\!\left(c-8\right)}\) that has no grouping symbols.
35.
Use the distributive property to write an equivalent expression to \({-\left(a+1\right)}\) that has no grouping symbols.
36.
Use the distributive property to write an equivalent expression to \({-\left(q+9\right)}\) that has no grouping symbols.
37.
Use the distributive property to simplify \({7+5\!\left(5+7m\right)}\) completely.
38.
Use the distributive property to simplify \({4+9\!\left(3+7p\right)}\) completely.
39.
Use the distributive property to simplify \({10-9\!\left(8+2q\right)}\) completely.
40.
Use the distributive property to simplify \({7-5\!\left(2+2y\right)}\) completely.
41.
Use the distributive property to simplify \({4-\left(-10-4r\right)}\) completely.
42.
Use the distributive property to simplify \({10-\left(-1-10a\right)}\) completely.
43.
Use the distributive property to simplify \({7-\left(5c+8\right)}\) completely.
44.
Use the distributive property to simplify \({4-\left(-9t+1\right)}\) completely.
45.
Use the distributive property to simplify \({{\frac{8}{3}}\!\left(1-4p\right)}\) completely.
46.
Use the distributive property to simplify \({{\frac{3}{5}}\!\left(-9-2m\right)}\) completely.
47.
Use the distributive property to simplify \({{\frac{3}{7}}\!\left(1+{\frac{6}{7}}p\right)}\) completely.
48.
Use the distributive property to simplify \({{\frac{9}{10}}\!\left(-3+{\frac{6}{7}}q\right)}\) completely.
49.
The expression \({y+c+b}\) would be ambiguous if we did not have a left-to-right reading convention. Use grouping symbols to emphasize the order that these additions should be carried out.
Use the associative property of addition to write an equivalent (but different) algebraic expression.
50.
The expression \({r+y+n}\) would be ambiguous if we did not have a left-to-right reading convention. Use grouping symbols to emphasize the order that these additions should be carried out.
Use the associative property of addition to write an equivalent (but different) algebraic expression.
51.
A student has (correctly) simplified an algebraic expression in the following steps. Between each pair of steps, identify the algebraic property that justifies moving from one step to the next.
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
52.
A student has (correctly) simplified an algebraic expression in the following steps. Between each pair of steps, identify the algebraic property that justifies moving from one step to the next.
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
commutative property of multiplication
associative property of addition
associative property of multiplication
distributive property
53.
The number of students enrolled in math courses at Portland Community College has grown over the years. The formulas
describe the numbers (of thousands) of men, women, and gender-non-binary students enrolled in math courses at PCC \(x\) years after 2005. (Note this is an exercise using randomized numbers, not actual data.) Give a simplified formula for the total number \(T\) of thousands of students at PCC taking math classes \(x\) years after 2005. Be sure to give the entire formula, starting with T=.
54.
The number of students enrolled in math courses at Portland Community College has grown over the years. The formulas
describe the numbers (of thousands) of men, women, and gender-non-binary students enrolled in math courses at PCC \(x\) years after 2005. (Note this is an exercise using randomized numbers, not actual data.) Give a simplified formula for the total number \(T\) of thousands of students at PCC taking math classes \(x\) years after 2005. Be sure to give the entire formula, starting with T=.