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.

 Combine like terms. Divide both sides by 12 to isolate x. Simplify. Check your answer. Let $$x=1$$

Example

Solve: $$11-20=17y-8y-6y.$$

Solution

Simplify each side by combining like terms.

 Simplify each side. Divide both sides by 3 to isolate y. Simplify. Check your answer. Let $$y=-3$$

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.

 Distribute. Simplify. Divide both sides by -3 to isolate n. Check your answer. Let $$n=-7$$.

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 $$ac=bc$$.