Simplifying Expressions - Math Steps, Examples & Questions (2024)

Introduction

What is simplifying expressions?

Common Core State Standards

How to simplify expressions

Teaching tips for simplifying expressions

Easy mistakes to make

Related algebraic expressions lessons

Practice simplifying expressions questions

Simplifying expressions FAQs

Next lessons

Still stuck?

Here you will learn about simplifying expressions, including using the distributive property and combining like terms.

Students will first learn about simplifying expressions as part of expressions and equations in 6th grade.

What is simplifying expressions?

Simplifying expressions involves using the properties of operations to create equivalent algebraic expressions.

For example,

Simplify 4(x+7)-x.

First, look at 4(x+7) which is ‘4 times the sum of x and 7.’

Notice that each method shows 4(x+7)=4x+28.

Since 4(x+7)-x=4x+28-x, you can continue to simplify by combining the like terms, in this case the x s.

Use the commutative property to change the order:

4x+28-x=4x-x+28

Next, let’s simplify the 4x-x .

Visually, if you have 4 positive x ‘s and we subtract an x, you are left with 3 positive x ‘s:

So, 4x-x=3x.

This means that 4x+28-x can be simplified to 3x+28.

You cannot simplify any further because ‘3x’ and ‘+28’ are not like terms. One has the variable x, and the other is a constant.

Common Core State Standards

How does this relate to 6th grade math?

• Grade 6 – Expressions and Equations (6.EE.A.3)
  Apply the properties of operations to generate equivalent expressions.
  For example, apply the distributive property to the expression 3 \, (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression \, 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

How to simplify expressions

In order to simplify expressions:

1. Multiply the terms within the parentheses by the term on the outside.
2. Combine like terms.
3. Write the simplified expression.

Example 1: simplify in one-step

Simplify 3 \, (5 + p).

1. Multiply the terms within the parentheses by the term on the outside.

3 \, (5 + p) is ‘3 times the sum of 5 and p.’

There are 3 different ways to solve:

Notice that each method shows 3 \, (5 + p) = 15 + 3p.

2Combine like terms.

You cannot simplify any further because ‘+3p’ and ‘15’’ are not like terms. One has the variable p, and the other is a constant.

3Write the simplified expression.

3 \, (5 + p) simplified is 15 + 3p.

Example 2: simplify in two-steps

Simplify 2 \, (2m + 6) + 1.

Multiply the terms within the parentheses by the term on the outside.

2 \, (2m + 6) is ‘2 times the sum of 2m and 6.’

There are 3 different ways to solve:

Example 3: simplify in one-step

Simplify 4 \, (8-k) + 3k.

Multiply the terms within the parentheses by the term on the outside.

4 \, (8-k) is ‘4 times the difference of 8 and k.’

There are 3 different ways to solve:

Combine like terms.

Since ‘-4k’ and ‘+3k’ are like terms, you can simplify further and combine them: -4k + 3k.

Visually if you have 4 negative ks and 3 positive ks, each negative and positive create a zero pair, leaving 1 negative k :

So, -4k + 3k = -k.

Example 4: simplify two-steps with the distributive property

Simplify 9 \, (1 + 2y) - 5y.

Multiply the terms within the parentheses by the term on the outside.

9 \, (1 + 2y) is ‘9 times the sum of 1 and 2y.’

Distribute the 9 to the 1 and the +2y.

So, 9 \, (1 + 2y)-5y = 9 + 18y-5y.

Combine like terms.

You can simplify further because ‘+18y’ and ‘-5y’ are like terms: 18y-5y.

Visually if you have 18 positive y s and then subtract 5 \, y s, there are 13 positive y s left:

So, 18y-5y = 13y.

Example 5: simplify two-steps with two variables with the distributive property

Simplify 7 \, (h + 4t)-9t.

Multiply the terms within the parentheses by the term on the outside.

7 \, (h + 4t) is ‘7 times the sum of h and +4t.’

Distribute the 7 to the h and the 4t.

So, 7\, (h + 4t)-9t = 7h + 28t-9t.

Example 6: perimeter multi-step with distributive property

Write the simplified expression for the perimeter of the shape.

Combine like terms.

You can simplify further because ‘+2x’ and ‘+4x’ are like terms: 2x + 4x = 6x.

‘-4’ and ‘6’ are also like terms: -4 + 6.

Visually if you have 4 negatives and add 6 positives, each negative and positive create a zero pair, leaving 2 positives:

So, -4 + 6 = 2.

Teaching tips for simplifying expressions

• Before students can simplify an algebraic expression, they need to understand exactly what makes up the expression. Encourage students to model expressions that have variables with hands-on manipulatives (such as algebra tiles), digital resources or their own drawings before beginning to simplify.

• Even if students have not learned how to operate with integers, they can still be successful at simplifying expressions like these. Encourage students to use what they know about positive and negative numbers and support their understanding with models, like those shown on this page.

Easy mistakes to make

• Confusing the order of operations (pemdas) when simplifying algebraic expressions
  Expressions should always be simplified following the order of operations. For most simple expressions, that means simplifying the parentheses using the distributive property and then adding the like terms from left to right. Doing this out of order, leads to the incorrect answer.

  For example,
  Simplify 7 \, (x +1)-4x +2x…

  

• Thinking the original expression and simplified are not equivalent
  Just like simplified fractions are equivalent versions of the original fractions, the same is true for simplified algebraic expressions. Using the distributive property and combining like terms does not change the value of the expression.

• Not including a positive or negative sign
  When combining the like terms, you must include the sign in front of the term.
  For example,
  5h + 8p-3h simplified is 2h + 8p.
  To combine the like h terms, solve 5h-3h. The subtraction or negative sign stays with the 3h. And the addition or positive sign stays with the 8h.

• Terms with a coefficient of \bf{1}
  For terms with a coefficient of 1, you don’t need to write the 1.
  For example,

  1x=x \hspace{0.6cm} 1y+ 4m=y + 4m \hspace{0.6cm} 6 \, (4 + 1r) = 6 \, (4 + r)

• Forgetting to use parentheses
  When multiplying an expression by a value, use parentheses so that each term is multiplied. Not using parentheses changes the outcome.
  For example,

  2 \, (y+4) = 2y + 8 \hspace{1.3cm} 2 \times y+4=2 y+4

Related algebraic expressions lessons

• Expanding expressions
• Equivalent expressions
• Evaluate the expression
• Polynomial

Is there a way to undo the simplification of an expression?

Can other types of algebraic expressions be simplified?

Can the distributive property be used only with multiplication?

The next lessons are

• Math equations
• Inequalities
• Math formulas
• Types of graphs

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