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Solve: 5 x = 4 x + 7 .

Solution

Here the variable, x , is on both sides, but the constants appear only on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.

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We don't want any variables on the right, so subtract the 4 x . .
Simplify. .
We have all the variables on one side and the constants on the other. We have solved the equation.
Check: .
Substitute 7 for x . .
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Solve: 6 n = 5 n + 10 .

n = 10

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Solve: −6 c = −7 c + 1 .

c = 1

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Solve: 5 y 8 = 7 y .

Solution

The only constant, −8 , is on the left side of the equation and variable, y , is on both sides. Let’s leave the constant on the left and collect the variables to the right.

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Subtract 5 y from both sides. .
Simplify. .
We have the variables on the right and the constants on the left. Divide both sides by 2. .
Simplify. .
Rewrite with the variable on the left. .
Check: Let y = −4 .
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Solve: 3 p 14 = 5 p .

p = −7

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Solve: 8 m + 9 = 5 m .

m = −3

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Solve: 7 x = x + 24 .

Solution

The only constant, 24 , is on the right, so let the left side be the variable side.

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Remove the x from the right side by adding x to both sides. .
Simplify. .
All the variables are on the left and the constants are on the right. Divide both sides by 8. .
Simplify. .
Check: Substitute x = 3 .
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Solve: 12 j = −4 j + 32 .

j = 2

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Solve: 8 h = −4 h + 12 .

h = 1

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Solve equations with variables and constants on both sides

The next example will be the first to have variables and constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side.

Solve: 7 x + 5 = 6 x + 2 .

Solution

Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are 7 x and 6 x . Since 7 is greater than 6 , make the left side the variable side and so the right side will be the constant side.

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Collect the variable terms to the left side by subtracting 6 x from both sides. .
Simplify. .
Now, collect the constants to the right side by subtracting 5 from both sides. .
Simplify. .
The solution is x = −3 .
Check: Let x = −3 .
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Solve: 12 x + 8 = 6 x + 2 .

x = −1

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Solve: 9 y + 4 = 7 y + 12 .

y = 4

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We’ll summarize the steps we took so you can easily refer to them.

Solve an equation with variables and constants on both sides.

  1. Choose one side to be the variable side and then the other will be the constant side.
  2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
  3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
  4. Make the coefficient of the variable 1 , using the Multiplication or Division Property of Equality.
  5. Check the solution by substituting it into the original equation.

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

Solve: 6 n 2 = −3 n + 7 .

Solution

We have 6 n on the left and −3 n on the right. Since 6 > 3 , make the left side the “variable” side.

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We don't want variables on the right side—add 3 n to both sides to leave only constants on the right. .
Combine like terms. .
We don't want any constants on the left side, so add 2 to both sides. .
Simplify. .
The variable term is on the left and the constant term is on the right.
To get the coefficient of n to be one, divide both sides by 9.
.
Simplify. .
Check: Substitute 1 for n .
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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