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You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that simplification is different from the process used to solve an equation in which we apply an operation to both sides.

How to solve equations that require simplification

Solve: 9 x 5 8 x 6 = 7 .

Solution

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Simplify the expressions on each side as much as possible.” The text in the second cell reads: “Rearrange the terms, using the Commutative Property of Addition. Combine like terms. Notice that each side is now simplified as much as possible.” The third cell contains the equation 9 x minus 5 minus 8 x minus 6 equals 7. Below this is the same equation, with the terms rearranged: 9 x minus 8 x minus 5 minus 6 equals 7. Below this is the equation with like terms combined: x minus 11 equals 7. In the second row of the table, the first cell says “Step 2. Isolate the variable.” In the second cell, the instructions say “Now isolate x. Undo subtraction by adding 11 to both sides.” The third cell contains the equation x minus 11 plus 11 equals 7 plus 11, with “plus 11” written in red on both sides. In the third row of the table, the first cell says: “Step 3. Simplify the equation on both sides of the equation.” The second cell is left blank. The third cell contains x equals 18. In the fourth and bottom row of the table, the first cell says: “Step 4. Check the solution.” The second cell is blank. In the third cell is the text “Check: Substitute x equals 18.” Below this is the equation 9 x minus 5 minus 8 x minus 6 equals 7. Underneath is the same equation, with 18 written in red in parentheses replacing each x: 9 times 18 (in parentheses) minus 5 minus 8 times 18 (in parentheses) minus 6 might equal 7. Below is the equation 162 minus 5 minus 144 minus 6 might equal 7. Below this is the equation 157 minus 144 minus 6 might equal 7. Below this is 13 minus 6 might equal 7. On the last line is the equation 7 equals 7, with a check mark next to it.
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Solve: 8 y 4 7 y 7 = 4 .

y = 15

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Solve: 6 z + 5 5 z 4 = 3 .

z = 2

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Solve: 5 ( n 4 ) 4 n = −8 .

Solution

We simplify both sides of the equation as much as possible before we try to isolate the variable.

.
Distribute on the left. .
Use the Commutative Property to rearrange terms. .
Combine like terms. .
Each side is as simplified as possible. Next, isolate n .
Undo subtraction by using the Addition Property of Equality. .
Add. .
Check. Substitute n = 12 .
.
The solution to 5 ( n 4 ) 4 n = −8 is n = 12 .
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Solve: 5 ( p 3 ) 4 p = −10 .

p = 5

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Solve: 4 ( q + 2 ) 3 q = −8 .

q = −16

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Solve: 3 ( 2 y 1 ) 5 y = 2 ( y + 1 ) 2 ( y + 3 ) .

Solution

We simplify both sides of the equation before we isolate the variable.

.
Distribute on both sides. .
Use the Commutative Property of Addition. .
Combine like terms. .
Each side is as simplified as possible. Next, isolate y .
Undo subtraction by using the Addition Property of Equality. .
Add. .
Check. Let y = −1 .
.
The solution to 3 ( 2 y 1 ) 5 y = 2 ( y + 1 ) 2 ( y + 3 ) is y = −1 .

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Solve: 4 ( 2 h 3 ) 7 h = 6 ( h 2 ) 6 ( h 1 ) .

h = 6

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Solve: 2 ( 5 x + 2 ) 9 x = 3 ( x 2 ) 3 ( x 4 ) .

x = 2

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Translate to an equation and solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign . [link] shows us some of the words that are commonly used.

Equals =
is
is equal to
is the same as
the result is
gives
was
will be

The steps we use to translate a sentence into an equation are listed below.

Translate an english sentence to an algebraic equation.

  1. Locate the “equals” word(s). Translate to an equals sign (=).
  2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
  3. Translate the words to the right of the “equals” word(s) into an algebraic expression.

Translate and solve: Eleven more than x is equal to 54.

Solution

Translate. .
Subtract 11 from both sides. .
Simplify. .
Check: Is 54 eleven more than 43?
43 + 11 = ? 54 54 = 54

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Translate and solve: Ten more than x is equal to 41.

x + 10 = 41 ; x = 31

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Translate and solve: Twelve less than x is equal to 51.

y 12 = 51 ; y = 63

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Translate and solve: The difference of 12 t and 11 t is −14 .

Solution

Translate. .
Simplify. .
Check:
12 ( −14 ) 11 ( −14 ) = ? −14 −168 + 154 = ? −14 −14 = −14
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Translate and solve: The difference of 4 x and 3 x is 14.

4 x 3 x = 14 ; x = 14

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Translate and solve: The difference of 7 a and 6 a is −8 .

7 a 6 a = −8 ; a = −8

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Translate and solve applications

Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.

We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q for the number of quarters if you were solving a problem about coins.

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?
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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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
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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
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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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?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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