Lesson 2: Evaluating Algebraic Expressions
Lesson Overview:
This lesson will introduce you to two important concepts:
- Substituting Values for Variables – You will learn how to replace variables in an expression with specific numbers and calculate the result.
- Order of Operations (PEMDAS) – You will practice solving algebraic expressions by following the correct order of operations: Parentheses, Exponents, Multiplication/Division (from left to right), and Addition/Subtraction (from left to right).
You can work through a PDF Worksheet to reinforce your learning at the end of the lesson.
Substituting Values for Variables:
Substituting values for variables is the process of replacing a variable in an algebraic expression with a specific number. This allows us to evaluate the expression and find a numerical result.
Example 1:
Expression: \(\style{font-size:10px}{2+3+x}\)
If \(\style{font-size:10px}{x=4}\), substitute \(\style{font-size:20px}4\) for \(\style{font-size:20px}x\)
\(\style{font-size:10px}{2+3+4=9}\)
Question 1:
Substitute \(\style{font-size:10px}{x=6}\) into the expression \(\style{font-size:10px}{3-2+x}\)
Solution:
\(\style{font-size:10px}{3-2+x}\)
\(\style{font-size:10px}{x=6}\)
\(\style{font-size:10px}{3-2+6=7}\)
Order of Operations (PEMDAS)
PEMDAS is a rule to follow when solving expressions with more than one operation. The acronym stands for:
- P: Parentheses
- E: Exponents (powers and square roots, etc.)
- MD: Multiplication and Division (from left to right)
- AS: Addition and Subtraction (from left to right)
Let’s have a look at a few examples of solving expressions using PEMDAS.
Example 2.1
Solve the expression: \(\style{font-size:10px}{3+5\times{(2+3)}^2}\)
Steps:
- Parentheses: \(\style{font-size:10px}{2+3=5}\)
- Exponent: \(\style{font-size:10px}{5^2=25}\)
- Multiplication: \(\style{font-size:10px}{5\times25=125}\)
- Addition: \(\style{font-size:10px}{3+125=128}\)
Example 2.2
Solve the expression: \(\style{font-size:10px}{12\div3+2\times4}\)
Steps:
- Division: \(\style{font-size:10px}{12\div3=4}\)
- Multiplication: \(\style{font-size:10px}{2\times4=8}\)
- Addition: \(\style{font-size:10px}{4+8=12}\)
Now that we understand how to apply the rules of PEMDAS, let’s practice some questions of solving algebraic expressions with more than one operation by substituting values.
Question 2:
Evaluate \(\style{font-size:10px}{3x^2+2y-z}\) when \(\style{font-size:10px}{x=2,\;y=4\;and\;z=5}\):
Solution:
- Substitute \(\style{font-size:10px}{x=2,\;y=4\;and\;z=5:}\) $$\style{font-size:10px}{3\left(2\right)^2+2(4)-5}$$
- Step 1 (Exponent): \(\style{font-size:10px}{2^2=4}\) $$\style{font-size:10px}{3(4)+2(4)-5}$$
- Step 2 (Multiplication): \(\style{font-size:10px}{3(4)=12,\;2(4)=8}\) $$\style{font-size:10px}{12+8-5}$$
- Step 3 (Addition/Subtraction): \(\style{font-size:10px}{12+8=20,\;20-5=15}\) $$\style{font-size:20px}{15}$$
Question 3:
Evaluate \(\style{font-size:10px}{5{(a+b)}^2\div c}\) when \(\style{font-size:10px}{a=3,\;b=2\;and\;c=5}\):
Solution:
- Substitute \(\style{font-size:10px}{a=3,\;b=2\;and\;c=5}\): $$\style{font-size:10px}{5{(3+2)}^2\div5}$$
- Step 1 (Parentheses): \(\style{font-size:10px}{3+2=5}\) $$\style{font-size:10px}{5{(5)}^2\div5}$$
- Step 2 (Exponent): \(\style{font-size:10px}{5^2=25}\) $$\style{font-size:10px}{5(25)\div5}$$
- Step 3 (Multiplication): \(\style{font-size:10px}{5(25)=125}\) $$\style{font-size:10px}{125\div5}$$
- Step 4 (Division): \(\style{font-size:10px}{125\div5=25}\) $$\style{font-size:20px}{25}$$
In this lesson, you explored the process of substituting values for variables and how to evaluate algebraic expressions using the order of operations, also known as PEMDAS. You learned how to replace variables with specific numbers and apply the proper sequence of operations (parentheses, exponents, multiplication/division, and addition/subtraction) to simplify expressions. These skills are essential for solving equations accurately and lay a strong foundation for more advanced problem-solving in algebra.
You should now have the knowledge of the basic algebraic rules. In the next lesson you will learn how to solve linear algebraic equations with one-step, two-step and multi-step equations with plenty of example and practice questions to help you understand solving algebraic equations.
Now that you’ve completed the lesson on evaluating algebraic expressions, give the worksheet a try to test your understanding and sharpen your skills!