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

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

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.

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

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

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

$-6m\text{=}-2$
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 $x-\text{14}+8=-2\text{.}$

$x=4$

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