Mathematics » Solving Linear Equations I » Solve Equations Using the Subtraction and Addition Properties of Equality

Solving Equations That Need to Be Simplified

Solving Equations That Need to Be Simplified

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

Example

Solve: \(3x-7-2x-4=1.\)

Solution

The left side of the equation has an expression that we should simplify before trying to isolate the variable.

 
 .
Rearrange the terms, using the Commutative Property of Addition..
Combine like terms..
Add 11 to both sides to isolate \(x\)..
Simplify..
Check.

 

Substitute \(x=12\) into the original equation.

 

The top line shows 3x minus 7 minus 2x minus 4 equals 1. Below this is 3 times a red 12 minus 7 minus 2 times a red 12 minus 4 equals 1. Next is 36 minus 7 minus 24 minus 4 equals 1. Below is 29 minus 24 minus 4 equals 1. Next is 5 minus 4 equals 1. Last is 1 equals 1.

 

The solution checks.

Example

Solve: \(3\left(n-4\right)-2n=-3.\)

Solution

The left side of the equation has an expression that we should simplify.

 .
Distribute on the left..
Use the Commutative Property to rearrange terms..
Combine like terms..
Isolate n using the Addition Property of Equality..
Simplify..
Check.

 

Substitute \(n=9\) into the original equation.

 

The top line says 3 times parentheses n minus 4 minus 2n equals negative 3. The next line says 3 times parentheses red 9 minus 3 minus 2 times red 9 equals negative 3. The next line says 3 times 5 minus 18 equals negative 3. Below this is 15 minus 18 equals negative 3. Last is negative 3 equals negative 3.

 

The solution checks.

 

Example

Solve: \(2\left(3k-1\right)-5k=-2-7.\)

Solution

Both sides of the equation have expressions that we should simplify before we isolate the variable.

 .
Distribute on the left, subtract on the right..
Use the Commutative Property of Addition..
Combine like terms..
Undo subtraction by using the Addition Property of Equality..
Simplify..
Check.

 

Let \(k=-7.\)

 

The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. Below this is 2 times parentheses red negative 7 minus 1 minus 5 times red negative 7 equals negative 2 minus 7. The next line says 2 times parentheses negative 21 minus 1 minus 5 times negative 7 equals negative 9. Below that is 2 times negative 22 plus 35 equals negative 9. Next is negative 44 plus 35 equals negative 9. The last line says negative 9 equals negative 9.

 

The solution checks.

 

Optional Video: Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Left)

Optional Video: Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Right)

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