Like Terms and Equivalent Expressions | Grade 7 Math | LevelKite

You already know how to write and evaluate expressions. Now you will learn to rewrite an expression in a shorter form that has the same value no matter what number the variable stands for.

Let $x$ stand for any number. The expression $3x$ means $3 \cdot x$ , which is three copies of that number added together: $x + x + x$ . So $3x + 5x$ is three copies of $x$ plus five more copies of $x$ . That makes eight copies in all, or $8x$ .

Now look at $3x + 5$ . This is three copies of $x$ plus the number $5$ . The $5$ is not a count of copies of $x$ , so it cannot join that count.

The rest of this lesson explains which parts of an expression you can combine, why combining them keeps the value the same, and how to do it step by step.

A term is a part of an expression that you add to the other parts. With $x$ as any number, the expression $3x + 5$ has two terms: $3x$ and $5$ .

In the term $3x$ , the number $3$ is the coefficient: the number that multiplies the variable. A term with no variable, such as $5$ , is a constant.

Two details matter when you list terms:

Subtracting a number gives the same result as adding its opposite. The opposite of $7$ is $-7$ , so $4x - 7$ means $4x + (-7)$ , and its terms are $4x$ and $-7$ . Keep each term's sign with it.
A variable standing alone has coefficient $1$ , because $1 \cdot x = x$ for every number $x$ . In the same way, $-x$ means $-1 \cdot x$ , so its coefficient is $-1$ .

Let $x$ , $y$ and $n$ stand for any numbers. A variable term in this lesson is a number times a variable, such as $3x$ or $-2y$ . Its variable part is that variable.

Like terms are terms with exactly the same variable part. A constant has no variable part, and constants are like terms with each other.

$3x$ and $-2x$ are like terms. Both have the variable part $x$ , so both are amounts of $x$ .
$3x$ and $3y$ are not like terms. The coefficients match, but the variable parts are different.
$4$ and $-9$ are like terms. Both are constants.

To decide, compare only the variable parts. The coefficients do not have to match.

Are the two terms like terms?

Terms	Variable parts	Like terms?	Reason
$7x$ and $2x$	$x$ and $x$	Yes	Same variable part. The coefficients may differ.
$4y$ and $-9y$	$y$ and $y$	Yes	Same variable part. A negative coefficient does not change the variable part.
$\frac{1}{2}n$ and $n$	$n$ and $n$	Yes	Same variable part. The term $n$ has coefficient $1$ .
$6$ and $-1.5$	None and none	Yes	Both are constants.
$5x$ and $5y$	$x$ and $y$	No	Different variable parts, even though the coefficients match.
$8x$ and $8$	$x$ and none	No	One term has a variable part and the other is a constant.

Worked example

Find the like terms

Let $x$ and $y$ stand for any numbers. Name the like terms in the expression $6x - 4 + 3y - x + 9$ .

Write each subtraction as adding the opposite, so that every term carries its own sign. Subtracting a number and adding its opposite give the same result for all numbers, so the value of the expression does not change.
$6x + (-4) + 3y + (-x) + 9$
List the terms, which are the parts you add. There are five of them.
$6x, \quad {-4}, \quad 3y, \quad {-x}, \quad 9$
Find the terms whose variable part is x. The term negative x means negative 1 times x, so it is an x term with coefficient negative 1.
$6x \text{ and } {-x}$
Find the terms whose variable part is y. Only one term has it, so that term has no like term in this expression.
$3y$
Find the constants, which are the terms with no variable. Constants are like terms with each other.
$-4 \text{ and } 9$

Answer

The like terms are $6x$ and $-x$ , and the constants $-4$ and $9$ are also like terms. The term $3y$ has no like term in this expression.

Two expressions are equivalent only if they have the same value for every value of the variable, not just for some values. In this lesson, an equals sign between two expressions with a variable means that they are equivalent. You are not solving for the variable. It can be any number, and the two expressions must agree every time.

You already know how to evaluate an expression by substituting a number for the variable. Substituting can show that two expressions are not equivalent: one value that gives two different results is enough. But a match for one value does not prove that they are equivalent, because equivalent means equal for every value and you cannot test them all.

To be sure, you use the properties of operations. Each property is true for all numbers. So when you rewrite an expression by using a property, the new expression has the same value as the old one for every value of the variable.

In the table below, $a$ , $b$ , $c$ and $x$ each stand for any number.

Properties of operations

Property	Rule	What it lets you do	Example
Commutative property of addition	$a + b = b + a$	Add in any order.	$3x + 7 = 7 + 3x$
Associative property of addition	$(a + b) + c = a + (b + c)$	Group the numbers you add in any way.	$(2x + 5) + 4 = 2x + (5 + 4)$
Commutative property of multiplication	$a \cdot b = b \cdot a$	Multiply in any order.	$x \cdot 6 = 6 \cdot x = 6x$
Associative property of multiplication	$(a \cdot b) \cdot c = a \cdot (b \cdot c)$	Group the numbers you multiply in any way.	$2 \cdot (3x) = (2 \cdot 3)x = 6x$
Distributive property	$a(b + c) = ab + ac$	Multiply a sum by multiplying each of its terms.	$2(3x + 4) = 2 \cdot 3x + 2 \cdot 4 = 6x + 8$

The distributive property also works from right to left: $ab + ac = a(b + c)$ . When two terms have the same factor, meaning the same number or variable multiplied in each, you can write that factor once and add what is left. You can multiply in either order, so you may also write the shared factor on the right. With $x$ as any number, this is why the eight copies of $x$ at the start of the lesson were correct:

3x + 5x = (3 + 5)x = 8x

Key concept

Why you can combine like terms

Like variable terms share a factor: the same variable. The distributive property lets you add their coefficients and keep the variable. The rule below is true for any numbers $a$ , $b$ and $x$ , including negative numbers, fractions and decimals. So its two sides are equivalent: they are equal for every value of $x$ .

Constants are like terms too. You combine them by adding the numbers.

A variable term and a constant are not like terms, and neither are two terms with different variables. The rule does not apply to them, so they stay as separate terms.

ax + bx = (a + b)x

Worked example

Use properties to write an equivalent expression

Let $x$ stand for any number. Use the properties of operations to write $2(x + 5) + 3x$ as an equivalent expression with only two terms.

The first term is 2 times a sum, so start with the distributive property: multiply 2 by each term inside the parentheses. The property is true for all numbers, so the value stays the same for every value of x.
$2(x + 5) + 3x = 2 \cdot x + 2 \cdot 5 + 3x = 2x + 10 + 3x$
Reorder so the like terms 2x and 3x are next to each other. You are adding all three terms, so the commutative and associative properties of addition allow any order and grouping.
$2x + 10 + 3x = 2x + 3x + 10$
Use the distributive property from right to left. The like terms 2x and 3x share the factor x, so write x once and add the coefficients.
$2x + 3x + 10 = (2 + 3)x + 10$
Add the coefficients. The terms 5x and 10 are not like terms, because one has the variable part x and the other is a constant. They stay separate.
$(2 + 3)x + 10 = 5x + 10$
Check for slips by substituting 4 for x in the first and last expressions. Both give 30. One matching value does not prove that the expressions are equivalent. The properties used in steps 1 to 4 do that, because they are true for all numbers.
$2(4 + 5) + 3 \cdot 4 = 2 \cdot 9 + 12 = 18 + 12 = 30 \quad \text{and} \quad 5 \cdot 4 + 10 = 20 + 10 = 30$

Answer

The expression $5x + 10$ is equivalent to $2(x + 5) + 3x$ . The two expressions have the same value for every value of $x$ .

In this lesson, to simplify an expression means to write an equivalent expression that has no parentheses and no like terms left to combine. When an expression has no parentheses, four moves will do it:

Write each subtraction as adding the opposite, so each term keeps its sign.
Reorder the terms so like terms are together. The commutative and associative properties of addition allow this.
For each group of like variable terms, add the coefficients and keep the variable. The distributive property allows this. Add the constants as numbers.
Check that no two of the remaining terms are like terms.

Moves 1 to 3 each use a fact that is true for all numbers. So the simplified expression is equal to the original one for every value of the variable.

Watch out for adding unlike terms. A common mistake is to rewrite $3x + 5$ as $8x$ , because $3 + 5 = 8$ .

Test it. Let $x$ stand for any number, and substitute $2$ for $x$ . The first expression gives $3 \cdot 2 + 5 = 11$ , but the second gives $8 \cdot 2 = 16$ . One value with two different results shows that the expressions are not equivalent. They do match when you substitute $1$ , because both give $8$ . That is why one match is not proof.

Why it fails: adding coefficients comes from the distributive property, and that needs both terms to have the same variable part. The term $3x$ has the variable part $x$ , but the constant $5$ has none. So the rule $ax + bx = (a + b)x$ does not apply, and $3x + 5$ is already simplified.

Worked example

Simplify with two variables and negative terms

Let $x$ and $y$ stand for any numbers. Simplify $8x - 3y - 5x + 7 + y - 10$ .

Write each subtraction as adding the opposite, so every term keeps its sign. This gives the same result for all numbers.
$8x + (-3y) + (-5x) + 7 + y + (-10)$
Reorder so like terms are together. You are adding all six terms, so the commutative and associative properties of addition allow the new order.
$8x + (-5x) + (-3y) + y + 7 + (-10)$
Combine the x terms with the distributive property: add the coefficients 8 and negative 5, and keep x.
$8x + (-5x) = (8 + (-5))x = 3x$
Combine the y terms the same way. The term y has coefficient 1, so add negative 3 and 1, and keep y.
$-3y + y = (-3 + 1)y = -2y$
The constants are like terms, so add them as numbers.
$7 + (-10) = -3$
Put the three results together, and write adding a negative as subtracting. The terms that remain are an x term, a y term and a constant, so no two are like terms and the expression is simplified.
$3x + (-2y) + (-3) = 3x - 2y - 3$

Answer

The simplified expression is $3x - 2y - 3$ . It is equal to $8x - 3y - 5x + 7 + y - 10$ for every value of $x$ and every value of $y$ .

Worked example

Simplify with fraction and decimal numbers

Let $n$ stand for any number. Simplify $\frac{3}{4}n - 2.5 + \frac{1}{2}n + 6 - n$ .

Write each subtraction as adding the opposite, which gives the same result for all numbers.
$\frac{3}{4}n + (-2.5) + \frac{1}{2}n + 6 + (-n)$
Reorder so like terms are together. Addition allows any order and grouping.
$\frac{3}{4}n + \frac{1}{2}n + (-n) + (-2.5) + 6$
Combine the three n terms with the distributive property: add their coefficients and keep n. The term negative n has coefficient negative 1. Using the two-term rule twice gives the same result.
$\frac{3}{4}n + \frac{1}{2}n + (-n) = \left(\frac{3}{4} + \frac{1}{2} + (-1)\right)n$
Add the coefficients. Rewrite them with the common denominator 4 first, so the parts are the same size.
$\frac{3}{4} + \frac{2}{4} + \left(-\frac{4}{4}\right) = \frac{5}{4} + \left(-\frac{4}{4}\right) = \frac{1}{4}$
The constants are like terms, so add them as numbers.
$-2.5 + 6 = 3.5$
Write the result. One term has the variable part n and the other is a constant, so they are not like terms and the expression is simplified.
$\frac{1}{4}n + 3.5$

Answer

The simplified expression is $\frac{1}{4}n + 3.5$ . It is equal to $\frac{3}{4}n - 2.5 + \frac{1}{2}n + 6 - n$ for every value of $n$ .

Here is the idea to keep. Like terms have exactly the same variable part, and constants are like terms with each other. The properties of operations are true for all numbers, so rewriting an expression with them gives an equivalent expression: one that is equal to the original for every value of the variable. Combining like terms is the distributive property at work: add the coefficients and keep the variable. A variable term and a constant, or two terms with different variables, stay separate.

Next, you will use these ideas to add and subtract linear expressions, which are expressions built from constants and terms that are a number times a variable.