# 2.1 Algebraic expressions

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## Coefficient

The coefficient of a quantity records how many of that quantity there are.

Since constants alone do not record the number of some quantity, they are not usually considered as numerical coefficients. For example, in the expression $7x+2y-8z+\text{12}$ , the coefficient of

$7x$ is 7. (There are 7 x 's.)
$2y$ is 2. (There are 2 y 's.)
$-8z$ is $-8$ . (There are $-8$ z 's.)

The constant 12 is not considered a numerical coefficient.

## $1x=x$

When the numerical coefficient of a variable is 1, we write only the variable and not the coefficient. For example, we write $x$ rather than $1x$ . It is clear just by looking at $x$ that there is only one.

## Numerical evaluation

We know that a variable represents an unknown quantity. Therefore, any expres­sion that contains a variable represents an unknown quantity. For example, if the value of $x$ is unknown, then the value of $3x+5$ is unknown. The value of $3x+5$ depends on the value of $x$ .

## Numerical evaluation

Numerical evaluation is the process of determining the numerical value of an algebraic expression by replacing the variables in the expression with specified numbers.

## Sample set b

Find the value of each expression.

$2x+7y$ , if $x=-4$ and $y=2$

Replace x with –4 and y with 2.

$\begin{array}{ccc}\hfill 2x+7y& =& 2\left(-4\right)+7\left(2\right)\hfill \\ & =& -8+14\hfill \\ & =& 6\hfill \end{array}$

Thus, when $x=–4$ and $y=2$ , $2x+7y=6$ .

$\frac{5a}{b}+\frac{8b}{12}$ , if $a=6$ and $b$ = $-3$ .

Replace a with 6 and b with –3.

$\begin{array}{ccc}\hfill \frac{5a}{b}+\frac{8b}{12}& =& \frac{5\left(6\right)}{-3}+\frac{8\left(-3\right)}{12}\hfill \\ & =& \frac{30}{-}+\frac{-}{24}\hfill \\ & =& -10+\left(-2\right)\hfill \\ & =& -12\hfill \end{array}$

Thus, when a = 6 and b = –3, $\frac{5a}{b}+\frac{8b}{\text{12}}=-\text{12}$ .

$6\left(2a-\text{15}b\right)$ , if $a=-5$ and $b=-1$

Replace $a$ with –5 and $b$ with –1.

$\begin{array}{ccc}\hfill 6\left(2a-15b\right)& =& 6\left(2\left(-5\right)-15\left(-1\right)\right)\hfill \\ & =& 6\left(-10+15\right)\hfill \\ & =& 6\left(5\right)\hfill \\ & =& 30\hfill \end{array}$

Thus, when $a=–5$ and $b=–1$ , $6\left(2a-15b\right)=30$ .

${3x}^{2}-2x+1$ , if $x=4$

Replace x with 4.

$\begin{array}{ccc}\hfill 3{x}^{2}-2x+1& =& 3{\left(4\right)}^{2}-2\left(4\right)+1\hfill \\ & =& 3\cdot 16-2\left(4\right)+1\hfill \\ & =& 48-8+1\hfill \\ & =& 41\hfill \end{array}$

Thus, when $x=4$ , ${3x}^{2}-2x+1=\text{41}$ .

$-{x}^{2}-4$ , if $x=3$

Replace x with 3.

${\left(-x\right)}^{2}-4$ , if $x=3$ .

Replace x with 3.

$\begin{array}{cccc}\hfill {\left(-x\right)}^{2}-4& =& {\left(-3\right)}^{2}-4\hfill & \text{The exponent is connected to -3, not 3 as in problem 5 above.}\hfill \\ & =& 9-4\hfill \\ & =& -5\hfill \end{array}$

The exponent is connected to –3, not 3 as in the problem above.

## Practice set b

Find the value of each expression.

$9m-2n,$ if $m=-2$ and $n=5$

-28

$-3x-5y+2z$ , if $x=-4$ , $y=3$ , $z=0$

-3

$\frac{\text{10}a}{3b}+\frac{4b}{2}$ , if $a=-6$ , and $b=2$

-6

$8\left(3m-5n\right)$ , if $m=-4$ and $n=-5$

104

$3\left[-\text{40}-2\left(4a-3b\right)\right]$ , if $a=-6$ and $b=0$

24

${5y}^{2}+6y-\text{11}$ , if $y=-1$

-12

$-{x}^{2}+2x+7$ , if $x=4$

-1

${\left(-x\right)}^{2}+2x+7$ , if $x=4$

31

## Exercises

In an algebraic expression, terms are separated by signs and factors are separated by signs.

For the following 8 problems, specify each term.

$3m+7n$

$5x+\text{18}y$

$5x,18y$

$4a-6b+c$

$8s+2r-7t$

$8s,2r,-7t$

$m-3n-4a+7b$

$7a-2b-3c-4d$

$7a,-2b,-3c,-4d$

$-6a-5b$

$-x-y$

$-x,-y$

What is the function of a numerical coefficient?

Write $1m$ in a simpler way.

$m$

Write 1 s in a simpler way.

In the expression 5 a , how many a ’s are indicated?

5

In the expression –7 c , how many c ’s are indicated?

Find the value of each expression.

$2m-6n$ , if $m=-3$ and $n=4$

-30

$5a+6b$ , if $a=-6$ and $b=5$

$2x-3y+4z$ , if $x=1$ , $y=-1$ , and $z=-2$

-3

$9a+6b-8x+4y$ , if $a=-2$ , $b=-1$ , $x=-2$ , and $y=0$

$\frac{8x}{3y}+\frac{\text{18}y}{2x},$ if $x=9$ and $y=-2$

-14

$\frac{-3m}{2n}-\frac{-6n}{m},$ if $m=-6$ and $n=3$

$4\left(3r+2s\right)$ , if $r=4$ and $s=1$

56

$3\left(9a-6b\right)$ , if $a=-1$ and $b=-2$

$-8\left(5m+8n\right)$ , if $m=0$ and $n=-1$

64

$-2\left(-6x+y-2z\right)$ , if $x=1$ , $y=1$ , and $z=2$

$-\left(\text{10}x-2y+5z\right)$ if $x=2$ , $y=8$ , and $z=-1$

1

$-\left(a-3b+2c-d\right)$ , if $a=-5$ , $b=2$ , $c=0$ , and $d=-1$

$3\left[\text{16}-3\left(a+3b\right)\right]$ , if $a=3$ and $b=-2$

75

$-2\left[5a+2b\left(b-6\right)\right]$ , if $a=-2$ and $b=3$

$-\left\{6x+3y\left[-2\left(x+4y\right)\right]\right\}$ , if $x=0$ and $y=1$

24

$-2\left\{\text{19}-6\left[4-2\left(a-b-7\right)\right]\right\}$ , if $a=\text{10}$ and $b=3$

${x}^{2}+3x-1$ , if $x=5$

39

${m}^{2}-2m+6$ , if $m=3$

${6a}^{2}+2a-\text{15}$ , if $a=-2$

5

${5s}^{2}+6s+\text{10},$ if $x=-1$

$\text{16}{x}^{2}+8x-7$ , if $x=0$

-7

$-{8y}^{2}+6y+\text{11},$ if $y=0$

${\left(y-6\right)}^{2}+3\left(y-5\right)+4$ , if $y=5$

5

${\left(x+8\right)}^{2}+4\left(x+9\right)+1,$ if $x=-6$

## Exercises for review

( [link] ) Perform the addition: $5\frac{3}{8}+2\frac{1}{6}$ .

$\frac{\text{181}}{\text{24}}=7\frac{\text{13}}{\text{24}}$

( [link] ) Arrange the numbers in order from smallest to largest: $\frac{\text{11}}{\text{32}},\frac{\text{15}}{\text{48}}\text{, and}\frac{7}{\text{16}}$

( [link] ) Find the value of ${\left(\frac{2}{3}\right)}^{2}+\frac{8}{\text{27}}$

$\frac{\text{20}}{\text{27}}$

( [link] ) Write the proportion in fractional form: “9 is to 8 as x is to 7.”

( [link] ) Find the value of $-3\left(2-6\right)-\text{12}$

0

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