Introduction to Multivariable Expressions
The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.
—JosephLouis Lagrange (1736–1813) Italian Mathematician and Astronomer
In this lesson, you'll learn how to simplify and evaluate multivariable expressions.
The previous lesson (SingleVariable Expressions Study Guide) showed us that we do not need to simplify singlevariable expressions. We can just substitute the value for the variable and evaluate. The same is true for multivariable expressions (expressions with more than one different variable), but you might find yourself performing many calculations if you do not simplify first.
To simplify a multivariable expression, we add, subtract, multiply, or divide all of the like terms until we are left with only unlike terms. Sometimes, we might take three, four, five, or even more steps to simplify an expression. Other times, there might not be a single operation we can perform. This occurs when all of the terms are unlike right from the start.
Example
Simplify 3x^{2} + 5y + 2x^{2} – 3y.
This expression has two x^{2} terms and two y terms. We can simplify the expression into just two terms by combining the two x^{2} terms and the two y terms: 3x^{2} + 2x^{2} = 5x^{2} and 5y – 3y = 2y. Therefore, 3x^{2} + 5y+ 2x^{2} – 3y simplifies to 5x^{2} + 2y.
Example
Simplify 2a^{3} – 4a + 5 + 9a – 6b.
There is only one a^{3} term, only one b term, and only one constant (5). None of these terms can be combined with any other terms. The only terms that can be combined are the a terms, –4a and 9a; –4a + 9a = 5a, so 2a^{3} – 4a + 5 + 9a – 6b simplifies to 2a^{3} + 5 + 5a – 6b. Remember, 2a^{3} and 5a are unlike terms. Although they have the same base, those bases have different exponents.
Example
Simplify 4w – 6t + 2s.
These terms are all unlike, because they all have different bases. This expression is already in its simplest form.
Tip:
When you think you have finished simplifying an expression, write all the terms that contain the same variable next to each other to be sure that there are no terms left that can be combined.

Example
Simplify –7gh + 8g – 2h + 5gh.
This expression contains three kinds of terms: terms with a base of g, terms with a base of h, and terms with a base of gh. The two terms with a base of gh can be combined: –7gh + 5gh = –2gh. Then, –7gh + 8g – 2h + 5gh simplifies to –2gh + 8g – 2h.
Example
Simplify 6x^{2} + 2x(x – 4y) + 3xy.
At first glance, this expression appears to contain four kinds of terms: x^{2} terms, x terms, y terms, and xy terms. However, the first step in simplifying this expression is to multiply: 2x(x – 4y) can be simplified using the distributive law. The distributive law says that a(b + c) = ab + ac. In words, the law says that the term outside the parentheses should be multiplied by each term inside the parentheses. To find 2x(x – 4y), multiply 2x by x and by –4y: 2x(x) = 2x^{2} and 2x(–4y) = –8xy. Our expression is now 6x^{2} + 2x^{2} – 8xy + 3xy. This expression has two x^{2} terms that can be combined and two xy terms that can be combined: 6x ^{2} + 2x^{2} = 8x^{2} and –8xy + 3xy = –5xy. The expression 6x^{2} + 2x(x – 4y) + 3xy simplifies to 8x^{2} –5xy.
Evaluating Multivariable Expressions
Now that we know how to simplify multivariable expressions, we can evaluate them. To evaluate a multivariable expression, replace each variable in the expression with its value. As always, remember the order of operations!
Example
What is 5y – 2z when y = 3 and z = –1?
The terms 5y and –2z are unlike, so they cannot be simplified. Replace y with 3 and replace z with –1:
5(3) – 2(–1) = 15 + 2 = 17
Example
What is 3a + 4b – 4a + 3b when a = 5 and b = 6?
First, combine the two a terms and the two b terms:
3a – 4a = –a
4b + 3b = 7b
The expression is now –a + 7b. Replace a with 5 and b with 6:
–5 + 7(6) = –5 + 42 = 37
Example
What is 3(s + 2t) – 6t + 4s when s = –2 and t = 32?
First, use the distributive law to find 3(s + 2t). Multiply 3 by s and multiply 3 by 2t:
3(s + 2t) = 3s + 6t
The expression is now:
3s + 6t – 6t + 4s
Combine the two s terms and the two t terms:
3s + 4s = 7s
6t – 6t = 0
The expression is now just 7s. Replace s with –2. There are no t terms to replace.
7(–2) = –14
The preceding example shows why simplifying an expression can often make evaluating an expression easier. Rather than having to add, subtract, multiply, or divide with 32, the value of t, in the end, the only calculation we had to do after substituting the value of s was to multiply 7 by –2.
Find practice problems and solutions for these concepts at Multivariable Expressions Practice Questions.
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