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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.
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 $x$ .
Find the value of each expression.
$2x+7y$ , if $x=-4$ and $y=2$
Replace x with –4 and y with 2.
$\begin{array}{ccc}\hfill 2x+7y& =& 2(-4)+7\left(2\right)\hfill \\ & =& -8+14\hfill \\ & =& 6\hfill \end{array}$
Thus, when $x=\u20134$ 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.
$\begin{array}{ccc}\hfill \frac{5a}{b}+\frac{8b}{12}& =& \frac{5\left(6\right)}{-3}+\frac{8(-3)}{12}\hfill \\ & =& \frac{30}{-}+\frac{-}{24}\hfill \\ & =& -10+(-2)\hfill \\ & =& -12\hfill \end{array}$
Thus, when a = 6 and b = –3, $\frac{\mathrm{5a}}{b}+\frac{\mathrm{8b}}{\text{12}}=-\text{12}$ .
$6\left(2a-\text{15}b\right)$ , if $a=-5$ and $b=-1$
Replace $a$ with –5 and $b$ with –1.
$\begin{array}{ccc}\hfill 6(2a-15b)& =& 6\left(2\right(-5)-15(-1\left)\right)\hfill \\ & =& 6(-10+15)\hfill \\ & =& 6\left(5\right)\hfill \\ & =& 30\hfill \end{array}$
Thus, when $a=\mathrm{\u20135}$ and $b=\mathrm{\u20131}$ , $6\left(2a-15b\right)=30$ .
${3x}^{2}-2x+1$ , 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.
$\begin{array}{cccc}\hfill {-x}^{2}-4& =& {-}^{3}-4\hfill & \text{Be careful to square only the 3. The exponent 2 is connected}\mathit{\text{only}}\text{to 3, not -3}\hfill \\ & =& -9-4\hfill \\ & =& -13\hfill \end{array}$
${\left(-x\right)}^{2}-4$ , if $x=3$ .
Replace x with 3.
$\begin{array}{cccc}\hfill {(-x)}^{2}-4& =& {(-3)}^{2}-4\hfill & \text{The exponent is connected to -3, not 3 as in problem 5 above.}\hfill \\ & =& 9-4\hfill \\ & =& -5\hfill \end{array}$
The exponent is connected to –3, not 3 as in the problem above.
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
$\frac{\text{10}a}{\mathrm{3b}}+\frac{\mathrm{4b}}{2}$ , 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
In an algebraic expression, terms are separated by
Addition; multiplication
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
$\left(x+8\right)}^{2}+4\left(x+9\right)+\mathrm{1,$ if $x=-6$
( [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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