# 1.1 Real numbers: algebra essentials  (Page 8/35)

 Page 8 / 35

List the constants and variables for each algebraic expression.

1. $2\pi r\left(r+h\right)$
2. 2( L + W )
3. $4{y}^{3}+y$
Constants Variables
a. $2\pi r\left(r+h\right)$ $2,\pi$ $r,h$
b. 2(L + W) 2 L, W
c. $\text{\hspace{0.17em}}4{y}^{3}+y$ 4 $y$

## Evaluating an algebraic expression at different values

Evaluate the expression $\text{\hspace{0.17em}}2x-7\text{\hspace{0.17em}}$ for each value for x.

1. $\text{\hspace{0.17em}}x=0$
2. $\text{\hspace{0.17em}}x=1$
3. $\text{\hspace{0.17em}}x=\frac{1}{2}$
4. $\text{\hspace{0.17em}}x=-4$
1. Substitute 0 for $\text{\hspace{0.17em}}x.$
$\begin{array}{ccc}\hfill 2x-7& =& 2\left(0\right)-7\\ & =& 0-7\hfill \\ & =& -7\hfill \end{array}$
2. Substitute 1 for $\text{\hspace{0.17em}}x.$
$\begin{array}{ccc}2x-7& =& 2\left(1\right)-7\hfill \\ & =& 2-7\hfill \\ & =& -5\hfill \end{array}$
3. Substitute $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x.$
$\begin{array}{ccc}\hfill 2x-7& =& 2\left(\frac{1}{2}\right)-7\hfill \\ & =& 1-7\hfill \\ & =& -6\hfill \end{array}$
4. Substitute $\text{\hspace{0.17em}}-4\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x.$
$\begin{array}{ccc}\hfill 2x-7& =& 2\left(-4\right)-7\\ & =& -8-7\hfill \\ & =& -15\hfill \end{array}$

Evaluate the expression $\text{\hspace{0.17em}}11-3y\text{\hspace{0.17em}}$ for each value for y.

1. $\text{\hspace{0.17em}}y=2$
2. $\text{\hspace{0.17em}}y=0$
3. $\text{\hspace{0.17em}}y=\frac{2}{3}$
4. $\text{\hspace{0.17em}}y=-5$
1. 5;
2. 11;
3. 9;
4. 26

## Evaluating algebraic expressions

Evaluate each expression for the given values.

1. $\text{\hspace{0.17em}}x+5\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x=-5$
2. $\text{\hspace{0.17em}}\frac{t}{2t-1}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}t=10$
3. $\text{\hspace{0.17em}}\frac{4}{3}\pi {r}^{3}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}r=5$
4. $\text{\hspace{0.17em}}a+ab+b\text{\hspace{0.17em}}$ for $a=11,b=-8$
5. $\text{\hspace{0.17em}}\sqrt{2{m}^{3}{n}^{2}}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}m=2,n=3$
1. Substitute $\text{\hspace{0.17em}}-5\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x.$
$\begin{array}{ccc}\hfill x+5& =& \left(-5\right)+5\hfill \\ & =& 0\hfill \end{array}$
2. Substitute 10 for $\text{\hspace{0.17em}}t.$
$\begin{array}{ccc}\hfill \frac{t}{2t-1}& =& \frac{\left(10\right)}{2\left(10\right)-1}\hfill \\ & =& \frac{10}{20-1}\hfill \\ & =& \frac{10}{19}\hfill \end{array}$
3. Substitute 5 for $\text{\hspace{0.17em}}r.$
$\begin{array}{ccc}\hfill \frac{4}{3}\pi {r}^{3}& =& \frac{4}{3}\pi {\left(5\right)}^{3}\\ & =& \frac{4}{3}\pi \left(125\right)\hfill \\ & =& \frac{500}{3}\pi \hfill \end{array}$
4. Substitute 11 for $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and –8 for $\text{\hspace{0.17em}}b.$
$\begin{array}{ccc}\hfill a+ab+b& =& \left(11\right)+\left(11\right)\left(-8\right)+\left(-8\right)\\ & =& 11-88-8\hfill \\ & =& -85\hfill \end{array}$
5. Substitute 2 for $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and 3 for $\text{\hspace{0.17em}}n.$
$\begin{array}{ccc}\hfill \sqrt{2{m}^{3}{n}^{2}}& =& \sqrt{2{\left(2\right)}^{3}{\left(3\right)}^{2}}\hfill \\ & =& \sqrt{2\left(8\right)\left(9\right)}\hfill \\ & =& \sqrt{144}\hfill \\ & =& 12\hfill \end{array}$

Evaluate each expression for the given values.

1. $\text{\hspace{0.17em}}\frac{y+3}{y-3}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}y=5$
2. $\text{\hspace{0.17em}}7-2t\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}t=-2$
3. $\text{\hspace{0.17em}}\frac{1}{3}\pi {r}^{2}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}r=11$
4. $\text{\hspace{0.17em}}{\left({p}^{2}q\right)}^{3}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}p=-2,q=3$
5. $\text{\hspace{0.17em}}4\left(m-n\right)-5\left(n-m\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}m=\frac{2}{3},n=\frac{1}{3}$
1. 4;
2. 11;
3. $\text{\hspace{0.17em}}\frac{121}{3}\pi$ ;
4. 1728;
5. 3

## Formulas

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 $\text{\hspace{0.17em}}2x+1=7\text{\hspace{0.17em}}$ has the unique solution of 3  because when we substitute 3 for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the equation, we obtain the true statement $\text{\hspace{0.17em}}2\left(3\right)+1=7.$

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 $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ of a circle in terms of the radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ of the circle: $\text{\hspace{0.17em}}A=\pi {r}^{2}.\text{\hspace{0.17em}}$ For any value of $\text{\hspace{0.17em}}r,$ the area $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ can be found by evaluating the expression $\text{\hspace{0.17em}}\pi {r}^{2}.$

## Using a formula

A right circular cylinder with radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ and height $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ has the surface area $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ (in square units) given by the formula $\text{\hspace{0.17em}}S=2\pi r\left(r+h\right).\text{\hspace{0.17em}}$ See [link] . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of $\text{\hspace{0.17em}}\pi .$

Evaluate the expression $\text{\hspace{0.17em}}2\pi r\left(r+h\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}r=6\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h=9.$

$\begin{array}{ccc}\hfill S& =& 2\pi r\left(r+h\right)\hfill \\ & =& 2\pi \left(6\right)\left[\left(6\right)+\left(9\right)\right]\hfill \\ & =& 2\pi \left(6\right)\left(15\right)\hfill \\ & =& 180\pi \hfill \end{array}$

The surface area is $\text{\hspace{0.17em}}180\pi \text{\hspace{0.17em}}$ square inches.

A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm 2 ) is found to be $\text{\hspace{0.17em}}A=\left(L+16\right)\left(W+16\right)-L\cdot W.\text{\hspace{0.17em}}$ See [link] . Find the area of a matte for a photograph with length 32 cm and width 24 cm.

1,152 cm 2

## Simplifying algebraic expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

## Simplifying algebraic expressions

Simplify each algebraic expression.

1. $3x-2y+x-3y-7$
2. $2r-5\left(3-r\right)+4$
3. $\left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)$
4. $2mn-5m+3mn+n$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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