11.3 Solving equations of the form x+a=b and x-a=b  (Page 2/2)

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From this, we can suggest the addition/subtraction property of equality .
Given any equation,
1. We can obtain an equivalent equation by adding the same number to both sides of the equation.
2. We can obtain an equivalent equation by subtracting the same number from both sides of the equation.

The idea behind equation solving

The idea behind equation solving is to isolate the variable on one side of the equation. Signs of operation (+, -, ⋅,÷) are used to associate two numbers. For example, in the expression $5+3$ , the numbers 5 and 3 are associated by addition. An association can be undone by performing the opposite operation. The addition/subtraction property of equality can be used to undo an association that is made by addition or subtraction.

Subtraction is used to undo an addition.

Addition is used to undo a subtraction.

The procedure is illustrated in the problems of [link] .

Sample set b

Use the addition/subtraction property of equality to solve each equation.

$x+4=6$ .
4 is associated with $x$ by addition. Undo the association by subtracting 4 from both sides.

$\begin{array}{c}x+4-4=6-4\hfill \\ x+0=2\hfill \\ x=2\hfill \end{array}$

Check: When $x=2$ , $x+4$ becomes

The solution to $x+4=6$ is $x=2$ .

$m-8=5$ . 8 is associated with $m$ by subtraction. Undo the association by adding 8 to both sides.

$\begin{array}{c}m-8+8=5+8\hfill \\ m+0=\text{13}\hfill \\ m=\text{13}\hfill \end{array}$

Check: When $m=\text{13}$ ,

becomes

a true statement.

The solution to $m-8=5$ is $m=\text{13}$ .

$-3-5=y-2+8$ . Before we use the addition/subtraction property, we should simplify as much as possible.

$-3-5=y-2+8$

$-8=y+6$
6 is associated with $y$ by addition. Undo the association by subtracting 6 from both sides.

$\begin{array}{c}-8-6=y+6-6\hfill \\ -\text{14}=y+0\hfill \\ -\text{14}=y\hfill \end{array}$
This is equivalent to $y=-\text{14}$ .

Check: When $y=-\text{14}$ ,

$-3-5=y-2+8$

becomes
,
a true statement.

The solution to $-3-5=y-2+8$ is $y=-\text{14}$ .

$-5a+1+6a=-2$ . Begin by simplifying the left side of the equation.

$\underset{-5+6=1}{\underbrace{-5a+1+6a}}=-2$

$a+1=-2$ 1 is associated with $a$ by addition. Undo the association by subtracting 1 from both sides.

$\begin{array}{c}a+1-1=-2-1\hfill \\ a+0=-3\hfill \\ a=-3\hfill \end{array}$

Check: When $a=-3$ ,

$-5a+1+6a=-2$

becomes
,
a true statement.

The solution to $-5a+1+6a=-2$ is $a=-3$ .

$7k-4=6k+1$ . In this equation, the variable appears on both sides. We need to isolate it on one side. Although we can choose either side, it will be more convenient to choose the side with the larger coefficient. Since 8 is greater than 6, we’ll isolate $k$ on the left side.

$7k-4=6k+1$ Since $6k$ represents $+6k$ , subtract $6k$ from each side.

$\underset{7-6=1}{\underbrace{7k-4-6k}}=\underset{6-6=0}{\underbrace{6k+1-6k}}$

$k-4=1$ 4 is associated with $k$ by subtraction. Undo the association by adding 4 to both sides.

$\begin{array}{c}k-4+4=1+4\hfill \\ k=5\hfill \end{array}$

Check: When $k=5$ ,

$7k-4=6k+1$

becomes

a true statement.

The solution to $7k-4=6k+1$ is $k=5$ .

$-8+x=5$ . -8 is associated with $x$ by addition. Undo the by subtracting -8 from both sides. Subtracting -8 we get $-\left(-8\right)\text{=+}8$ . We actually add 8 to both sides.

$-8+x+8=5+8$

$x=\text{13}$

Check: When $x=\text{13}$

$-8+x=5$

becomes
,
a true statement.

The solution to $-8+x=5$ is $x=\text{13}$ .

Practice set b

$y+9=4$

$y=-5$

$a-4=\text{11}$

$a=\text{15}$

$-1+7=x+3$

$x=3$

$8m+4-7m=\left(-2\right)\left(-3\right)$

$m=2$

$\text{12}k-4=9k-6+2k$

$k=-2$

$-3+a=-4$

$a=-1$

Exercises

For the following 10 problems, verify that each given value is a solution to the given equation.

$x-\text{11}=5$ , $x=\text{16}$

Substitute $x=4$ into the equation $4x-\text{11}=5$ .
$\begin{array}{}\text{16}-\text{11}=5\\ 5=5\end{array}$
$x=4$ is a solution.

$y-4=-6$ , $y=-2$

$2m-1=1$ , $m=1$

Substitute $m=1$ into the equation $2m-1=1$ .

$m=1$ is a solution.

$5y+6=-\text{14}$ , $y=-4$

$3x+2-7x=-5x-6$ , $x=-8$

Substitute $x=-8$ into the equation $3x+2-7=-5x-6$ .

$x=-8$ is a solution.

$-6a+3+3a=4a+7-3a$ , $a=-1$

$-8+x=-8$ , $x=0$

Substitute $x=0$ into the equation $-8+x=-8$ .

$x=0$ is a solution.

$8b+6=6-5b$ , $b=0$

$4x-5=6x-\text{20}$ , $x=\frac{\text{15}}{2}$

Substitute $x=\frac{\text{15}}{2}$ into the equation $4x-5=6x-\text{20}$ .

$x=\frac{\text{15}}{2}$ is a solution.

$-3y+7=2y-\text{15}$ , $y=\frac{\text{22}}{5}$

Solve each equation. Be sure to check each result.

$y-6=5$

$y=\text{11}$

$m+8=4$

$k-1=4$

$k=5$

$h-9=1$

$a+5=-4$

$a=-9$

$b-7=-1$

$x+4-9=6$

$x=\text{11}$

$y-8+\text{10}=2$

$z+6=6$

$z=0$

$w-4=-4$

$x+7-9=6$

$x=8$

$y-2+5=4$

$m+3-8=-6+2$

$m=1$

$z+\text{10}-8=-8+\text{10}$

$2+9=k-8$

$k=\text{19}$

$-5+3=h-4$

$3m-4=2m+6$

$m=\text{10}$

$5a+6=4a-8$

$8b+6+2b=3b-7+6b-8$

$b=-\text{21}$

$\text{12}h-1-3-5h=2h+5h+3\left(-4\right)$

$-4a+5-2a=-3a-\text{11}-2a$

$a=\text{16}$

$-9n-2-6+5n=3n-\left(2\right)\left(-5\right)-6n$

Calculator exercises

$y-2\text{.}\text{161}=5\text{.}\text{063}$

$y=7\text{.}\text{224}$

$a-\text{44}\text{.}\text{0014}=-\text{21}\text{.}\text{1625}$

$-0\text{.}\text{362}-0\text{.}\text{416}=5\text{.}\text{63}m-4\text{.}\text{63}m$

$m=-0\text{.}\text{778}$

$8\text{.}\text{078}-9\text{.}\text{112}=2\text{.}\text{106}y-1\text{.}\text{106}y$

$4\text{.}\text{23}k+3\text{.}\text{18}=3\text{.}\text{23}k-5\text{.}\text{83}$

$k=-9\text{.}\text{01}$

$6\text{.}\text{1185}x-4\text{.}\text{0031}=5\text{.}\text{1185}x-0\text{.}\text{0058}$

$\text{21}\text{.}\text{63}y+\text{12}\text{.}\text{40}-5\text{.}\text{09}y=6\text{.}\text{11}y-\text{15}\text{.}\text{66}+9\text{.}\text{43}y$

$y=-\text{28}\text{.}\text{06}$

$0\text{.}\text{029}a-0\text{.}\text{013}-0\text{.}\text{034}-0\text{.}\text{057}=-0\text{.}\text{038}+0\text{.}\text{56}+1\text{.}\text{01}a$

Exercises for review

( [link] ) Is $\frac{7\text{calculators}}{\text{12}\text{students}}$ an example of a ratio or a rate?

rate

( [link] ) Convert $\frac{3}{8}\text{}$ % to a decimal.

( [link] ) 0.4% of what number is 0.014?

3.5

( [link] ) Use the clustering method to estimate the sum: $\text{89}+\text{93}+\text{206}+\text{198}+\text{91}$

( [link] ) Combine like terms: $4x+8y+\text{12}y+9x-2y$ .

$\text{13}x+\text{18}y$

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