# 11.4 Solving equations of the form ax=b and x/a=b  (Page 2/2)

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$\frac{3x}{8}=6$

$x=\text{16}$

$-y=3$

$y=-3$

$-k=-2$

$k=2$

## Combining techniques in equation solving

Having examined solving equations using the addition/subtraction and the multi­plication/division principles of equality, we can combine these techniques to solve more complicated equations.

When beginning to solve an equation such as $6x-4=-\text{16}$ , it is helpful to know which property of equality to use first, addition/subtraction or multiplication/di­vision. Recalling that in equation solving we are trying to isolate the variable (disas­sociate numbers from it), it is helpful to note the following.

To associate numbers and letters, we use the order of operations.

1. Multiply/divide
2. Add/subtract

To undo an association between numbers and letters, we use the order of opera­tions in reverse.

1. Add/subtract
2. Multiply/divide

## Sample set b

Solve each equation. (In these example problems, we will not show the checks.)

$6x-4=-\text{16}$
-4 is associated with $x$ by subtraction. Undo the association by adding 4 to both sides.

$6x-4+4=-\text{16}+4$

$6x=-\text{12}$
6 is associated with $x$ by multiplication. Undo the association by dividing both sides by 6

$\frac{6x}{6}=\frac{-\text{12}}{6}$

$x=-2$

$-8k+3\text{=}-45\text{.}$
3 is associated with $k$ by addition. Undo the association by subtracting 3 from both sides.

$-8k+3-3\text{=}-45-3$

$-8k\text{=}-48$
-8 is associated with $k$ by multiplication. Undo the association by dividing both sides by -8.

$\frac{-8k}{-8}=\frac{-\text{48}}{-8}$

$k=6$

$5m-6-4m=4m-8+3m\text{.}$ Begin by solving this equation by combining like terms.

$m-6=7m-8$ Choose a side on which to isolate m . Since 7 is greater than 1, we'll isolate m on the right side.
Subtract m from both sides.

$-m-6-m=7m-8-m$

$-6=6m-8$
8 is associated with m by subtraction. Undo the association by adding 8 to both sides.

$-6+8=6m-8+8$

$2=6m$
6 is associated with m by multiplication. Undo the association by dividing both sides by 6.

$\frac{2}{6}=\frac{6m}{6}$ Reduce.

$\frac{1}{3}=m$

Notice that if we had chosen to isolate m on the left side of the equation rather than the right side, we would have proceeded as follows:

$m-6=7m-8$
Subtract $7m$ from both sides.

$m-6-7m=7m-8-7m$

$-6m-6\text{=}-8$
Add 6 to both sides,

$-6m-6+6\text{=}-8+6$

$-6m\text{=}-2$
Divide both sides by -6.

$\frac{-6m}{-6}=\frac{-2}{-6}$

$m=\frac{1}{3}$

This is the same result as with the previous approach.

$\frac{8x}{7}\text{=}-2$
7 is associated with $x$ by division. Undo the association by multiplying both sides by 7.

$\overline{)7}\cdot \frac{8x}{\overline{)7}}=7\left(-2\right)$

$7\cdot \frac{8x}{7}\text{=}-\text{14}$

$8x\text{=}-\text{14}$
8 is associated with $x$ by multiplication. Undo the association by dividing both sides by 8.

$\frac{\overline{)8}x}{\overline{)8}}=\frac{-7}{4}$

$x=\frac{-7}{4}$

## Practice set b

Solve each equation. Be sure to check each solution.

$5m+7\text{=}-\text{13}$

$m\text{=}-4$

$-3a-6=9$

$a\text{=}-5$

$2a+\text{10}-3a=9$

$a=1$

$\text{11}x-4-\text{13}x=4x+\text{14}$

$x\text{=}-3$

$-3m+8=-5m+1$

$m\text{=}-\frac{7}{2}$

$5y+8y-\text{11}\text{=}-\text{11}$

$y=0$

## Exercises

Solve each equation. Be sure to check each result.

$7x=\text{42}$

$x=6$

$8x=\text{81}$

$\text{10}x=\text{120}$

$x=\text{12}$

$\text{11}x=\text{121}$

$-6a=\text{48}$

$a\text{=}-8$

$-9y=\text{54}$

$-3y\text{=}-\text{42}$

$y=\text{14}$

$-5a\text{=}-\text{105}$

$2m\text{=}-\text{62}$

$m\text{=}-\text{31}$

$3m\text{=}-\text{54}$

$\frac{x}{4}=7$

$x=\text{28}$

$\frac{y}{3}=\text{11}$

$\frac{-z}{6}\text{=}-\text{14}$

$z=\text{84}$

$\frac{-w}{5}=1$

$3m-1\text{=}-\text{13}$

$m=-4$

$4x+7\text{=}-\text{17}$

$2+9x\text{=}-7$

$x=-1$

$5-\text{11}x=\text{27}$

$\text{32}=4y+6$

$y=\frac{\text{13}}{2}$

$-5+4=-8m+1$

$3k+6=5k+\text{10}$

$k\text{=}-2$

$4a+\text{16}=6a+8a+6$

$6x+5+2x-1=9x-3x+\text{15}$

$x=\frac{\text{11}}{2}\text{or 5}\frac{1}{2}$

$-9y-8+3y+7=-7y+8y-5y+9$

$-3a=a+5$

$a=-\frac{5}{4}$

$5b\text{=}-2b+8b+1$

$-3m+2-8m-4=-\text{14}m+m-4$

$m\text{=}-1$

$5a+3=3$

$7x+3x=0$

$x=0$

$7g+4-\text{11}g=-4g+1+g$

$\frac{5a}{7}=\text{10}$

$a=\text{14}$

$\frac{2m}{9}=4$

$\frac{3x}{4}=\frac{9}{2}$

$x=6$

$\frac{8k}{3}=\text{32}$

$\frac{3a}{8}-\frac{3}{2}=0$

$a=4$

$\frac{5m}{6}-\frac{\text{25}}{3}=0$

## Exercises for review

( [link] ) Use the distributive property to compute $\text{40}\cdot \text{28}$ .

$\text{40}\left(\text{30}-2\right)=\text{1200}-\text{80}=\text{1120}$

( [link] ) Approximating $\pi$ by 3.14, find the approximate circumference of the circle. ( [link] ) Find the area of the parallelogram. 220 sq cm

( [link] ) Find the value of $\frac{-3\left(4-\text{15}\right)-2}{-5}$ .

( [link] ) Solve the equation $x-\text{14}+8=-2\text{.}$

$x=4$

#### Questions & Answers

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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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CYNTHIA
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s. Reply
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of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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7hours 36 min - 4hours 50 min
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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