Evaluating expressions

On this post we explain how to evaluate algebraic expressions. Also, you will see examples of evaluations of expressions and you will find practice problems on evaluating algebraic expressions solved step by step.

How to evaluate expressions

To evaluate an algebraic expression for a given value, you have to substitute the variable of the expression for the value and perform all the arithmetic operations of the expression.

For example, to evaluate the expression 3x+5 for x=1, you have to replace the variable x with the number 1 and perform the calculations: 3(1)+5=8.

So that you can better understand the concept of the evaluation of an expression, below we evaluate an algebraic expression step by step.

  • What is the evaluation of the following algebraic expression for x=2?

x^2-4x+1

To find the evaluation of the expression, we have to substitute the variable x for the value x=2. Thus:

2^2-4\cdot 2+1

And once we have substituted the value in the algebraic expression, we compute all the operations. So first we solve the power:

4-4\cdot 2+1

Now we multiply:

4-8+1

And finally, we add and subtract the terms:

\bm{-3}

In conclusion, the evaluation of the expression for the value x=2 is equal to -3.

As you can see, evaluating expressions is not very complicated, however, it has very useful applications. For example, evaluating an algebraic expression is needed to use the remainder theorem, a very important theorem of polynomials.

What is the remainder theorem?

Examples of evaluations of expressions

So that you finish understanding how to evaluate an algebraic expression, below you have several examples:

Example 1

  • Evaluate the expression -2x^2+3x+5 when x=-1.

 \begin{array}{l}-2\cdot (-1)^2+3\cdot (-1)+5= \\[2ex]=-2\cdot 1+3\cdot (-1)+5 =\\[2ex]=-2-3+5 =\\[2ex]= \bm{0} \end{array}

Example 2

  • Evaluate the algebraic expression -2x^3+7x^2-8x-2 for x=3.

 \begin{array}{l}-2\cdot 3^3+7\cdot 3^2-8\cdot 3-2= \\[2ex]=-2\cdot 27+7\cdot 9-8\cdot 3-2= \\[2ex]=-54+63-24-2= \\[2ex]= \bm{-17} \end{array}

Evaluating expressions with two or more variables

We have just seen how to find the evaluation of an expression with only one variable. But… how do you evaluate an algebraic expression when it has more than one variable?

If an expression has 2 or more letters, you have to apply the same procedure, that is, first each variable of the expression is replaced by its corresponding value and then the arithmetic operations are performed.

As an example, you have an exercise solved below:

  • Evaluate the expression with two variables x^2y+4xy-2x^2+6y-5 for the values x=2,y=3.

First of all, we substitute each variable for its corresponding value, that is, we substitute the letter x for 2 and we substitute the letter y for 3:

2^2\cdot 3+4\cdot 2\cdot 3-2\cdot 2^2+6\cdot 3-5

We solve the powers:

4\cdot 3+4\cdot 2\cdot 3-2\cdot 4+6\cdot 3-5

Now we calculate the multiplications:

12+24-8+18-5

And finally, we combine terms:

\bm{41}

Practice problems on evaluating expressions

Problem 1

Evaluate the following expression for x=-2.

-2x^3-4x^2+3x+8

To find the value of the expression we simply have to substitute the given value in the expression and compute the operations:

\begin{array}{l}-2\cdot (-2)^3-4\cdot (-2)^2+3\cdot (-2)+8 =\\[2ex]=-2\cdot (-8)-4\cdot 4+3\cdot (-2)+8 =\\[2ex]=+16-16-6+8 =\\[2ex]= \bm{2} \end{array}

 

Problem 2

Evaluate the following algebraic expression with fractions for the value x=4.

\cfrac{1}{2} x^2-\cfrac{5}{4}x + 7

Regardless of whether the expression has fractions or not, the procedure to evaluate it is always the same. So we must replace the variable x with 4 and solve the calculations:

 \begin{array}{l}\cfrac{1}{2} \cdot 4^2-\cfrac{5}{4}\cdot 4+ 7 =\\[2ex] =\cfrac{1}{2} \cdot 16-\cfrac{5}{4}\cdot 4+ 7=\\[2ex] =8-5+7 =\\[2ex]= \bm{10} \end{array}

 

Problem 3

Evaluate the following expression with three variables for the values x=3, y=5 and z=-2.

 x^2yz+4y^2z^2-3x^2z+6xyz

To evaluate a multi-variable expression we just have to plug the values given in the algebraic expression and solve the resulting operations:

 \begin{array}{l}3^2\cdot 5\cdot (-2)+4\cdot 5^2\cdot (-2)^2-3\cdot 3^2\cdot (-2)+6\cdot 3\cdot 5 \cdot (-2) =\\[2ex]=9\cdot 5\cdot (-2)+4\cdot 25\cdot 4-3\cdot 9\cdot (-2)+6\cdot 3\cdot 5 \cdot (-2) =\\[2ex] =-90+400+54-180=\\[2ex]= \bm{184} \end{array}

 

Problem 4

Given the expression -2x^3-3x^2+5x+k, find the value of k so that evaluation of the expression for x=-2 equals to 5.

First we try to evaluate the algebraic expression when x=-2.

 \begin{array}{l}-2\cdot (-2)^3-3\cdot (-2)^2+5\cdot (-2)+k= \\[2ex] =-2\cdot (-8)-3\cdot 4+5\cdot (-2)+k =\\[2ex]=+16-12-10+k=\\[2ex]=-6+k \end{array}

The evaluation of the expression must be equal to 5. Therefore, we set the obtained expression equal to 5:

-6+k=5

And we solve the equation:

-6+k+6=5+6

 \bm{k=11}

 

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