Evaluating Expressions: Definition, Solved Examples

Evaluating Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator

In mathematics, algebraic expressions are constructed upon constants and variables. They are also made up of algebraic operations such as addition, subtraction, etc. Sometimes, these expressions can be seen in terms as well. 

Once we define these expressions, we have to evaluate them. Evaluating an expression includes finding a missing variable. This is by replacing it with a given number. 

Moreover, it does not just stop over there. One needs to translate and simplify these sentences as well. All these concepts, including how to evaluate expressions with a deep understanding of subtopics, will be explained below. 

How to Evaluate Expressions

What does “evaluate the expression” mean? Simplification of expressions can be done with the help of operations’ order. The following passage is about evaluating expressions. It includes replacing the variable to find the expression’s value. The given number is substituted to evaluate expressions. After this, one needs to simplify it based on the order of the operations. This is how we evaluate each expression.

Let us look at an example:

Example 1: Evaluate y + 12 when

1. y = 15
2. y = 3

Solution:

1. Substitute 15 for y in the given expression:

=y + 12

=15 + 12 = 27

Hence, the value of y is 27.

2. Substitute 3 for y in the given expression:

=y + 12

=3 + 12 = 15

Hence, the value of y is 15.

From the example, it is seen that it is possible to obtain various results for the same expression. It is because y had different values. So, from this, one can understand that the results completely rely on the value of the variable.

Let us look at more examples and understand how to evaluate the expression

Example 2: Evaluate expression: x + 3

1. x = 6
2. x = 9

Solution:

1. Substitute 6 for x in the given expression:

= x + 3

= 6 + 3 = 9

Hence, the value of x is 9.

2. Substitute 9 for x in the given expression:

= 9 + 3

= 9+3 = 12

Hence, the value of x is 12.

Let’s see some more complex examples, where we evaluate each expression:

Example 3: Evaluate expression: 2 z + 3, when

1. z = 2
2. z = 5

Solution:

To evaluate, one needs to keep in mind that 2 z means 2 times the value of z. 

1. Substitute 2 for z in the given expression:

2 z + 3

= 2 (2) + 3

= 4 + 3

= 7

Hence, the value of z is 7.

2. Substitute 5 for z in the given expression:

2 z + 3

= 2 (5) + 3

= 10 + 3

= 13

Hence, the value of z is 13.

Example 4: Evaluate expression: y² when y = 3

Solution:

Substituting 3 for y in the given expression:

y² = 3²

= 3 * 3

= 9

Hence, the value of y² is 9, while substituting y = 3.

Example 5: Evaluate expression: 3^Z when z = 4.

Solution:

Substituting 4 for z in the given expression:

=3^Z

= 3⁴

= 3 * 3 * 3 * 3

= 81

Hence, the value of 3^Z is 81, while substituting z = 4.

Example 6: Evaluate expression: 7 x + 5 y – 2 when x = 5 and y = 4

Solution: 

This example consists of two variables, to evaluate this expression, one should substitute both the values.

As this expression consists of two variables, one should substitute both values.

Substituting 5 for x and 4 for y in the given expression,

= 7 x + 5 y – 2

= 7 (5) + 5 (4) -2

= 35 + 20 -2

= 53

Hence, the value of the expression 7 x + 5 y – 2 is 53.

Lastly, let us see an example with an expression that contains a variable with an exponent.

Example 7: Evaluate expression: 2 y² + 3 y + 4 when y = 2

Solution:

The expression needs to be solved carefully. It is because the variable in it contains an exponent. In the expression, 2 y² indicates 2 * y * y. It is different from the expression, (2 y)², which denotes 2 y * 2 y.

Substitute 2 for each y in the given expression,

= 2 y² + 3 y + 4

= 2 (2)² + 3 (2) + 4

= 8 + 6 + 4

= 18

Hence, the value of the expression 2 y² + 3 y + 4 when y = 2 is 18.

Identifying terms, coefficients, and like terms

These expressions are also made up of terms. A term can be denoted as a constant. In other cases, terms can be products of constants and one or more variables. Examples of terms are 5, 6 x, y², 12 x y, etc. 

A coefficient is nothing but a constant that multiplies the variable. They are present in front of the variables. For example, in the term 4 y, the coefficient here is 4. But in the term x, the coefficient here is 1. 

Let us have a look at the following to know about terms and coefficients even better:

Term	Coefficient
14 x	14
25 z2	25
12	12
a2	1
x	1

Algebraic Expressions: Algebraic Expressions are used to calculate solutions for any Mathematical operations that include variables such as addition, subtraction, multiplication or division. There are three types of algebraic expressions; Monomial Expression, Binomial Expression, and Polynomial Expression.

An unknown value is represented as the letters x,y and z in the fundamentals of Algebraic expressions. These letters are referred to as variables. An algebraic expression can have both variables and constants. Together they form the algebraic expression.

Algebraic expressions are constructed up on one or more terms. For example:

Expression	Terms
9	9
z	z
y + 9	y, 9
5 x + 6 y + 16	5 x, 6 y, 16
6 z4 + 9 y + 14 x2 + 12 y  3	6 z4, 9 y, 14 x2, 12 y, 3
2 x2 + 7 y2 + 2 x y + 9	2 x2, 7 y2, 2 xy, 9

Here are a few more examples:

Example 8: Determine all the terms in the given expression.

14 a² + 2 b + 6 x y + 3. Also, find the coefficient of each term. 

Solution:

The given expression consists of four terms in it. They are 14 a² , 2 b, 6 x y, and 3.

The coefficient of 14 a² is 14.

The coefficient of 2 b is 2.

The coefficient of 6 x y is 6.

Finally, 3 is already a coefficient, since it is a constant.

Example 9: Determine all the terms and their coefficients in the given expression:

a³ + 9 y⁵ + 6 x² + 4.

Solution:

The given expression has four terms in it. They are a³, 9 y⁵, 6 x², 4.

The coefficients are 1, 9, 6, and 4.

Moreover, there are certain terms in an expression that could share common traits. For example:

4 y, 8, m², 2, 3 y, 6 m²

The following are the like terms for the given example:

• The terms m² and 6 m² are both terms with m². 
• The terms 8 and 2 are considered constant terms.
• The terms 4 y and 3 y are regarded as y terms.

From this, one can easily understand that if two or more terms have the same variables and exponents, then they are said to be like terms. In addition to this, all constant terms are like terms. So, due to this reason:

4 y and 3 y are like terms.

m² and 6 m² are like terms.

8 and 2 are like terms.

Looking at the following example:

Example 10: Determine the like terms:

x² + 4 y + 8 z³ + 7 + z³ + 2 y + 7 x² + 3 + 12 x

Solution:

While looking for similar terms, one needs to focus on the variables and exponents. The expression has x2 terms, y terms, z3 terms, and constant terms.

The terms x² and 7 x² are like terms. This is because they both have x² in them.

The terms 4 y and 2 y are like terms. This is because they both have y in them.

The terms 8 z³ and z³ are like terms. This is because they both have z³ in them.

The terms 7 and 3 are constants. So they are also like terms.

Lastly, the term 12 x does not have a similar term. So it is not a like term.

Simplifying expressions by combining like terms

This is a very simple concept. It just means adding the like terms together and forming a whole new term, which is a combination of both.

For example, let’s take the expression, 3 z + 4 z.

As discussed earlier, we just need to add both coefficients and keep the variable as it is.

It becomes, 3 z + 4 z = 7 z, which means 7 times the value of z.

Let us have a look at another example:

6 y + 7 z – 4 y +  z.

This expression becomes, 6 y –  4y + 7 z + z. It gives, 2 y + 8 z.

From this learners can easily understand that the expression becomes easier by:

• Identifying like terms.
• Keeping like terms next to each other by rearranging the expression.
• Adding or subtracting the coefficients of the like terms.

Conclusion

Expressions are the most commonly used items in mathematics. One needs to understand how to evaluate expressions to simplify the problems easily. What does it mean to evaluate an expression? All the students learning this chapter should have a detailed idea of that. In addition to this, it is essential to understand what evaluating the expression means. Everything mentioned above was deeply discussed in this chapter.