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3 x 8 = 6 size 12{ { {3x} over {8} } =6} {}

x = 16 size 12{x="16"} {}

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y = 3 size 12{-y=3} {}

y = - 3 size 12{y"=-"3} {}

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k = - 2 size 12{-k"=-"2} {}

k = 2 size 12{k=2} {}

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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 6 x - 4 = 16 size 12{6 ital "x-"4= - "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.)

6 x 4 = - 16 size 12{6x-4"=-""16"} {}
-4 is associated with x by subtraction. Undo the association by adding 4 to both sides.

6 x 4 + 4 = - 16 + 4 size 12{6x-4+4"=-""16"+4} {}

6 x = - 12 size 12{6x"=-""12"} {}
6 is associated with x by multiplication. Undo the association by dividing both sides by 6

6 x 6 = 12 6 size 12{ { {6x} over {6} } = { {-"12"} over {6} } } {}

x = - 2 size 12{x"=-"2} {}

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8 k + 3 = -45 . size 12{-8k+3"=-""45" "." } {}
3 is associated with k by addition. Undo the association by subtracting 3 from both sides.

8 k + 3 3 = -45 3 size 12{-8k+3-3"=-""45"-3} {}

8 k = - 48 size 12{-8k"=-""48"} {}
-8 is associated with k by multiplication. Undo the association by dividing both sides by -8.

8 k 8 = 48 8 size 12{ { {-8k} over {-8} } = { {-"48"} over {-8} } } {}

k = 6 size 12{k=6} {}

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5 m 6 4 m = 4 m 8 + 3 m . size 12{5m-6-4m=4m-8+3m "." } {} Begin by solving this equation by combining like terms.

m 6 = 7 m 8 size 12{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 = 7 m 8 m size 12{-m-6-m=7m-8-m} {}

6 = 6 m 8 size 12{-6=6m-8} {}
8 is associated with m by subtraction. Undo the association by adding 8 to both sides.

6 + 8 = 6 m 8 + 8 size 12{-6+8=6m-8+8} {}

2 = 6 m size 12{2=6m} {}
6 is associated with m by multiplication. Undo the association by dividing both sides by 6.

2 6 = 6 m 6 size 12{ { {2} over {6} } = { {6m} over {6} } } {} Reduce.

1 3 = m size 12{ { {1} over {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 = 7 m 8 size 12{m-6=7m-8} {}
Subtract 7 m from both sides.

m 6 7 m = 7 m 8 7 m size 12{m-6-7m=7m-8-7m} {}

6 m 6 = -8 size 12{-6m-6"=-"8} {}
Add 6 to both sides,

6 m 6 + 6 = -8 + 6 size 12{-6m-6+6"=-"8+6} {}

6 m = - 2 size 12{-6m"=-"2} {}
Divide both sides by -6.

6 m 6 = 2 6 size 12{ { {-6m} over {-6} } = { {-2} over {-6} } } {}

m = 1 3 size 12{m= { {1} over {3} } } {}

This is the same result as with the previous approach.

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8 x 7 = - 2 size 12{ { {8x} over {7} } "=-"2} {}
7 is associated with x by division. Undo the association by multiplying both sides by 7.

7 8 x 7 = 7 2 size 12{7 cdot { {8x} over {7} } =7 left (-2 right )} {}

7 8 x 7 = - 14 size 12{ { {7}} cdot { {8x} over { { {7}}} } "=-""14"} {}

8 x = - 14 size 12{8x"=-""14"} {}
8 is associated with x by multiplication. Undo the association by dividing both sides by 8.

8 x 8 = 7 4 size 12{ { { { {8}}x} over { { {8}}} } = { {-7} over {4} } } {}

x = 7 4 size 12{x= { {-7} over {4} } } {}

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Practice set b

Solve each equation. Be sure to check each solution.

5 m + 7 = - 13 size 12{5m+7"=-""13"} {}

m = - 4 size 12{m"=-"4} {}

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3 a 6 = 9 size 12{-3a-6=9} {}

a = - 5 size 12{a"=-"5} {}

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2 a + 10 3 a = 9 size 12{2a+"10"-3a=9} {}

a = 1 size 12{a=1} {}

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11 x 4 13 x = 4 x + 14 size 12{"11"x-4-"13"x=4x+"14"} {}

x = - 3 size 12{x"=-"3} {}

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3 m + 8 = - 5 m + 1 size 12{-3m+8"=-"5m+1} {}

m = - 7 2 size 12{m"=-" { {7} over {2} } } {}

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5 y + 8 y 11 = - 11 size 12{5y+8y-"11""=-""11"} {}

y = 0 size 12{y=0} {}

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Exercises

Solve each equation. Be sure to check each result.

7 x = 42 size 12{7x="42"} {}

x = 6 size 12{x=6} {}

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8 x = 81 size 12{8x="81"} {}

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10 x = 120 size 12{"10"x="120"} {}

x = 12 size 12{x="12"} {}

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11 x = 121 size 12{"11"x="121"} {}

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6 a = 48 size 12{-6a="48"} {}

a = - 8 size 12{a"=-"8} {}

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9 y = 54 size 12{-9y="54"} {}

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3 y = - 42 size 12{-3y"=-""42"} {}

y = 14 size 12{y="14"} {}

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5 a = - 105 size 12{-5a"=-""105"} {}

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2 m = - 62 size 12{2m"=-""62"} {}

m = - 31 size 12{m"=-""31"} {}

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3 m = - 54 size 12{3m"=-""54"} {}

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x 4 = 7 size 12{ { {x} over {4} } =7} {}

x = 28 size 12{x="28"} {}

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y 3 = 11 size 12{ { {y} over {3} } ="11"} {}

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z 6 = - 14 size 12{ { {-z} over {6} } "=-""14"} {}

z = 84 size 12{z="84"} {}

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w 5 = 1 size 12{ { {-w} over {5} } =1} {}

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3 m 1 = - 13 size 12{3m-1"=-""13"} {}

m = - 4 size 12{m"=-"4} {}

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4 x + 7 = - 17 size 12{4x+7"=-""17"} {}

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2 + 9 x = - 7 size 12{2+9x"=-"7} {}

x = - 1 size 12{x"=-"1} {}

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5 11 x = 27 size 12{5-"11"x="27"} {}

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32 = 4 y + 6 size 12{"32"=4y+6} {}

y = 13 2 size 12{y= { {"13"} over {2} } } {}

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5 + 4 = - 8 m + 1 size 12{-5+4"=-"8m+1} {}

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3 k + 6 = 5 k + 10 size 12{3k+6=5k+"10"} {}

k = - 2 size 12{k"=-"2} {}

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4 a + 16 = 6 a + 8 a + 6 size 12{4a+"16"=6a+8a+6} {}

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6 x + 5 + 2 x 1 = 9 x 3 x + 15 size 12{6x+5+2x-1=9x-3x+"15"} {}

x = 11 2 or 5 1 2 size 12{x= { {"11"} over {2} } " or 5" { {1} over {2} } } {}

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9 y 8 + 3 y + 7 = - 7 y + 8 y 5 y + 9 size 12{-9y-8+3y+7"=-"7y+8y-5y+9} {}

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3 a = a + 5 size 12{-3a=a+5} {}

a = - 5 4 size 12{a"=-" { {5} over {4} } } {}

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5 b = - 2 b + 8 b + 1 size 12{5b"=-"2b+8b+1} {}

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3 m + 2 8 m 4 = - 14 m + m 4 size 12{-3m+2-8m-4"=-""14"m+m-4} {}

m = - 1 size 12{m"=-"1} {}

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5 a + 3 = 3 size 12{5a+3=3} {}

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7 x + 3 x = 0 size 12{7x+3x=0} {}

x = 0 size 12{x=0} {}

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7 g + 4 11 g = - 4 g + 1 + g size 12{7g+4-"11"g"=-"4g+1+g} {}

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5 a 7 = 10 size 12{ { {5a} over {7} } ="10"} {}

a = 14 size 12{a="14"} {}

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2 m 9 = 4 size 12{ { {2m} over {9} } =4} {}

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3 x 4 = 9 2 size 12{ { {3x} over {4} } = { {9} over {2} } } {}

x = 6 size 12{x=6} {}

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8 k 3 = 32 size 12{ { {8k} over {3} } ="32"} {}

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3 a 8 3 2 = 0 size 12{ { {3a} over {8} } - { {3} over {2} } =0} {}

a = 4 size 12{a=4} {}

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5 m 6 25 3 = 0 size 12{ { {5m} over {6} } - { {"25"} over {3} } =0} {}

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Exercises for review

( [link] ) Use the distributive property to compute 40 28 size 12{"40" cdot "28"} {} .

40 30 2 = 1200 80 = 1120 size 12{"40" left ("30"-2 right )="1200"-"80"="1120"} {}

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( [link] ) Approximating π by 3.14, find the approximate circumference of the circle.

A circle with radius 8cm.

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( [link] ) Find the area of the parallelogram.

A parallelogram with base 20cm and height 11cm.

220 sq cm

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( [link] ) Find the value of 3 4 15 2 5 size 12{ { {-3 left (4-"15" right )-2} over {-5} } } {} .

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( [link] ) Solve the equation x 14 + 8 = - 2 . size 12{x-"14"+8"=-"2 "." } {}

x = 4 size 12{x=4} {}

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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