Simplifying a Formula

A rectangle with length $L$ and width $W$ has a perimeter $P$ given by $P=L+W+L+W$. Simplify this expression.

Solution

$$\begin{align*} P &= L+W+L+W &&\quad \\ P &= L+L+W+W &&\quad \text{Cummutative property of addition} \\ P &= 2L+2W &&\quad \text{Simplify} \\ P &= 2(L+W) &&\quad \text{Distributive property} \\ \end{align*}$$

Case

The above case, including the title and solution, is attributed to College Algebra 2e. Page 20-21.

Q&A

What are order of operations?

Page 12-13, College Algebra 2e

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS:

• P(arentheses)
• E(xponents)
• M(ultiplication) and D(ivision)
• A(ddition) and S(ubtraction)

What are commutative properties?

Page 14, College Algebra 2e

The commutative property of addition states that numbers may be added in any order without affecting the sum. $a+b=b+a$

We can better see this relationship when using real numbers. $(-2)+7=5 \quad \text{and} \quad 7+(-2)=5$

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product. $a \cdot b=b \cdot a$

Again, consider an example with real numbers. $(-11)\cdot(-4)=44 \quad \text{and} \quad (-4)\cdot(-11)=44$

It is important to note that neither subtraction nor division is commutative. For example, $17-5$ is not the same as $5-17$ Similarly, $20\div5 \not = 5\div20$.

What are associative properties?

Page 14-15, College Algebra 2e

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same. $a(bc)=(ab)c$

Consider this example. $(3\cdot4)\cdot5=60 \quad \text{and} \quad 3\cdot(4\cdot5)=60$

The associative property of addition tells us that numbers may be grouped differently without affecting the sum. $a+(b+c)=(a+b)+c$

This property can be especially helpful when dealing with negative integers. Consider this example. $[15+(-9)]+23=29 \quad \text{and} \quad 15+[(-9)+23]=29$

Are subtraction and division associative? Review these examples.

$$\begin{align*} {4} 8-(3-15) &\stackrel{?}{=} (8-3)-15 &\quad 64\div(8\div4) &\stackrel{?}{=} (64\div8)\div4 \\ 8-(-12) &= 5-15 &\quad 64\div2 &\stackrel{?}{=} 8\div4 \\ 20 &\not= -10 &\quad 32 &\not= 2 \end{align*}$$

As we can see, neither subtraction nor division is associative.

What are distributive properties?

Page 15, College Algebra 2e

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. $a \cdot (b+c) = a \cdot b + a \cdot c$

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

$$\begin{align*} 6+(3\cdot5) &\stackrel{?}{=} (6+3)\cdot(6+5) \\ 6+(15) &\stackrel{?}{=} (9)\cdot(11) \\ 21 &\not= 99 \end{align*}$$

A special case of the distributive property occurs when a sum of terms is subtracted. $a-b=a+(-b)$

For example, consider the difference $12-(5+3)$. We can rewrite the difference of the two terms $12$ and $(5+3)$ by turning the subtraction expression into addition of the opposite. So instead of subtracting $(5+3)$ we add the opposite. $12+(-1)\cdot(5+3)$

Now, distribute $-1$ and simplify the result.

$$\begin{align*} 12-(5+3) &= 12+(-1)\cdot(5+3) \\ &= 12+[(-1)\cdot5+(-1)\cdot3] \\ &= 12+(-8) \\ &= 4 \end{align*}$$

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

$$\begin{align*} 12-(5+3) &= 12+(-5-3) \\ &= 12+(-8) \\ &= 4 \end{align*}$$

What are identity properties?

Page 15, College Algebra 2e

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number. $a+0=a$

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number. $a\cdot1=a$

For example, we have $(-6)+0=-6$ and $23\cdot1=23$. There are no exceptions for these properties; they work for every real number, including $0$ and $1$.

What are inverse properties?

Page 16, College Algebra 2e

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted by $(−a)$, that, when added to the original number, results in the additive identity, $0$. $a+(-a)=0$

For example, if $a=-8$, the additive inverse is $8$, since $(-8)+8=0$.

The inverse property of multiplication holds for all real numbers except $0$ because the reciprocal of $0$ is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted $\frac{1}{a}$, that, when multiplied by the original number, results in the multiplicative identity, $1$. $a\cdot\frac{1}{a}=1$

For example, if $a=-\frac{2}{3}$, the reciprocal, denoted $\frac{1}{a}$, is $\frac{3}{2}$ because $a\cdot\frac{1}{a}=\Big(-\frac{2}{3}\Big)\cdot\Big(-\frac{3}{2}\Big)=1$

What are properties of real numbers?

Page 16, College Algebra 2e

The following properties hold for real numbers a, b, and c.

	Addition	Multiplication
Commutative Property	$a+b=b+a$	$a \cdot b=b \cdot a$
Associative Property	$a+(b+c)=(a+b)+c$	$a(bc)=(ab)c$
Distributive Property	$a\cdot(b+c)=a \cdot b+a \cdot c$	$a\cdot(b+c)=a \cdot b+a \cdot c$
Identity Property	There exists a unique real number called the additive identity, $0$, such that, for any real number $a$ $a+0=a$	There exists a unique real number called the multiplicative identity, $1$, such that, for any real number $a$ $a\cdot1=a$
Inverse Property	Every real number $a$ has an additive inverse, or opposite, denoted $-a$, such that $a+(-a)=0$	Every nonzero real number $a$ has a multiplicative inverse, or reciprocal, denoted $\frac{1}{a}$, such that $a \cdot \Big(\frac{1}{a}\Big)=1$

Evaluating Algebraic Expressions

Page 17, College Algebra 2e

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as $x+5,\frac{4}{3} \pi r^3$, or $\sqrt{2m^3n^2}$ . In the expression $x+5$, $5$ is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

$$\begin{align*} {3} (-3)^5 &= (-3)\cdot(-3)\cdot(-3)\cdot(-3)\cdot(-3) &\quad x^5 &= x \cdot x \cdot x \cdot x \cdot x \\ (2\cdot7)^3 &= (2\cdot7) \cdot (2\cdot7) \cdot (2\cdot7) &\quad (yz)^3 &= (yz)\cdot(yz)\cdot(yz) \\ \end{align*}$$

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

What is an equation?

Page 19, College Algebra 2e

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation $2x+1=7$ has the solution $3$ because when we substitute $3$ for $x$ in the equation, we obtain the true statement $2(3)+1=7$.

What is a formula?

Page 19, College Algebra 2e

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area $A$ of a circle in terms of the radius $r$ of the circle: $A=\pi r^2$. For any value of $r$, the area $A$ can be found by evaluating the expression $\pi r^2$.

Glossary

Associative Property

https://en.wikipedia.org/wiki/Associative_property

The associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result.

Consider the following equations:

$$\begin{aligned} (2+3)+4 &= 2+(3+4) = 9 \\ 2\times (3\times 4) &= (2\times 3)\times 4 = 24 \end{aligned}$$

Last accessed

January 01, 2024

Commutative Property

https://en.wikipedia.org/wiki/Commutative_property

A binary operation is commutative if changing the order of the operands does not change the result.

E.g. $3 + 4 = 4 + 3$ or $2 × 5 = 5 × 2$

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Distributive Property

https://en.wikipedia.org/wiki/Distributive_property

The distributive property of binary operations is a generalization of the distributive law, which asserts that the equality

$$x\cdot (y+z) = x\cdot y + x\cdot z$$

is always true in elementary algebra. For example, in elementary arithmetic, one has

$$2\cdot (1+3) = 2\cdot 1 + 2\cdot 3$$

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Equation

https://en.wikipedia.org/wiki/Equation

An equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign $=$.

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Expression

https://en.wikipedia.org/wiki/Expression_(mathematics)

An expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

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Identity Property

https://simple.wikipedia.org/wiki/Identity_Property

In math, the identity property is made up of two parts: the additive identity property and the multiplicative identity property.

The additive identity property says that the sum of adding any number and zero (0) is just the original number. For that reason, zero is often called the additive identity of common numbers.

The multiplicative identity property says that the product of multiplying any number and one (1) is just the original number. Also, if you divide a number by itself, the result (quotient) is one. For that reason, one is often called the multiplicative identity of common numbers.

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Inverse Property

https://en.wikipedia.org/wiki/Additive_inverse

https://en.wikipedia.org/wiki/Multiplicative_inverse

The additive inverse of a number $a$(sometimes called the opposite of $a$) is the number that, when added to $a$, yields zero.

A multiplicative inverse or reciprocal for a number $x$, denoted by $\frac{1}{x}$ or $x^{−1}$, is a number which when multiplied by $x$ yields the multiplicative identity, $1$

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Order of Operations

https://en.wikipedia.org/wiki/Order_of_operations

The order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.

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Perimeter

https://en.wikipedia.org/wiki/Perimeter

A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.

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note

• Content last updated on January 01, 2024
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