Mathematics » Solving Linear Equations I » Solve Equations Using the Division and Multiplication Properties of Equality

Solving Equations That Need to be Simplified

Solving Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

Example

Solve: \(8x+9x-5x=-3+15.\)

Solution

Start by combining like terms to simplify each side.

 Solving Equations That Need to be Simplified
Combine like terms.Solving Equations That Need to be Simplified
Divide both sides by 12 to isolate x.Solving Equations That Need to be Simplified
Simplify.Solving Equations That Need to be Simplified
Check your answer. Let \(x=1\) 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 

Example

Solve: \(11-20=17y-8y-6y.\)

Solution

Simplify each side by combining like terms.

 Solving Equations That Need to be Simplified
Simplify each side.Solving Equations That Need to be Simplified
Divide both sides by 3 to isolate y.Solving Equations That Need to be Simplified
Simplify.Solving Equations That Need to be Simplified
Check your answer. Let \(y=-3\) 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 

Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.

Example

Solve: \(-3\left(n-2\right)-6=21.\)

Solution

Remember—always simplify each side first.

 Solving Equations That Need to be Simplified
Distribute.Solving Equations That Need to be Simplified
Simplify.Solving Equations That Need to be Simplified
Divide both sides by -3 to isolate n.Solving Equations That Need to be Simplified

 

Solving Equations That Need to be Simplified

Check your answer. Let \(n=-7\). 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 
Solving Equations That Need to be Simplified 

Optional Video: Solving One Step Equation in the Form: −x = −a

Key Concepts

  • Division and Multiplication Properties of Equality
    • Division Property of Equality: For all real numbers a, b, c, and \(c\ne 0\), if \(a=b\), then \(ac=bc\).
    • Multiplication Property of Equality: For all real numbers a, b, c, if \(a=b\), then \(ac=bc\).

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This is a lesson from the tutorial, Solving Linear Equations I and you are encouraged to log in or register, so that you can track your progress.

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