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Evaluating Expressions: Definition, Solved Examples

evaluate expressions

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.

  1. 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 have a look at more examples to understand and evaluate each expression

Example 2: Evaluate 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.

  1. 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 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.

  1. 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 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 3Z when z = 4.

Solution:

Substituting 4 for z in the given expression:

=3Z

= 34

= 3 * 3 * 3 * 3

= 81

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

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

Solution: 

As this expression consists of two variables, one should substitute both the 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 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:

TermCoefficient 
14 x14
25 z225
1212
a21
x1

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

Expression Terms
99
zz
y + 9y, 9
5 x + 6 y + 165 x, 6 y, 16
6 z4 + 9 y + 14 x2 + 12 y  36 z4, 9 y, 14 x2, 12 y, 3
2 x2 + 7 y2 + 2 x y + 92 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 the 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 does evaluate the expression means. Everything mentioned above was deeply discussed in this chapter. 

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