Algebraic expressions are fundamental components of algebra that represent quantities and relationships using variables, constants, and operations. They form the basis for solving equations and understanding mathematical relationships.

In this article, we will learn what algebraic expressions are and solve problems related to them.

What is Algebraic expression?

An algebraic expression can be simple or complex and describes a wide range of mathematical phenomena.

The algebraic expression does not contain an equality sign. It simply represents a value that can be calculated or simplified based on the variables’ values.

Example:

• 5x + 3
• 2x² – 4x + 1
• [Tex]\frac{2x^2 + 3x – 5}{x – 1} [/Tex]
• [Tex]\sqrt{x^2 + 4}[/Tex], etc.

Distributive Property

Distributive property allows you to eliminate parentheses by distributing multiplication over addition or subtraction within the parentheses.

• Formula:

a(b+c) = ab + ac

Combining Like Terms

Combine terms with the same variable to simplify the equation.

• Formula:

ax + bx = (a+b)x

Important Concepts

Algebraic expression contains:

• Variables: Symbols (such as x, y, z) that represent unknown values or quantities.
• Constants: Fixed numerical values (such as 1, -5, 2.3) that do not change.
• Operators: Symbols that indicate operations to be performed, including addition (+), subtraction (-), multiplication (× or ·), and division (÷ or /).
• Exponents: Powers to which variables are raised (such as x² or y³).

How to Solve Algebraic Expression

To solve practice questions follow the steps added below:

Step 1: Identify Terms then combine them together.

Step 2: Distribute and Combine

Step 3: Identify Common Factors

Step 4: Isolate the Variable

Step 5: Simplify the results

Algebraic Expressions Practice Questions

Question 1: Simplify Algebraic Expression: 4(a+3) -2(2a-5) + 3(a+1).

Solution:

Distribute the 4, -2, and 3, then combine like terms.

= 4a + 12 – 4a + 10 + 3a + 3

= 3a + 25

Question 2: Simplify Algebraic Expression: 7x+4x.

Solution:

= 7x + 4x

= 11x

Question 3: Simplify Algebraic Expression: 3(x+2)+4x

Solution:

= 3(x+2) + 4x

= 3x + 6 + 4x

= 7x + 6

Question 4: Simplify Algebraic Expression: [Tex]\frac{5x^2 – 3x + 2}{x^2 + 2x + 1}​[/Tex]

Solution:

Factor denominator x² + 2x + 1

Recognize that x² + 2x + 1 = (x+1)²

So, we have:

[Tex]\frac{5x^2 – 3x + 2}{(x + 1)^2}[/Tex]

Numerator 5x²– 3x + 2 cannot be easily factored further in a way that will cancel out with the denominator. Therefore, this fraction is already in its simplest form.

Question 5: Simplify Algeabric Expression: -2(3y-4)

Solution:

= -2(3y-4)

= -6y+8

Question 6: Simplify Algeabric Expression: 5a-3+2a+7

Solution:

= 5a + 2a – 3 + 7

= 7a + 4

Question 7: Simplify Algeabric Expression: [Tex]\frac{(x – 1)^2}{x^2 – 2x + 1} + \frac{(x + 1)^2}{x^2 + 2x + 1}[/Tex]

Solution:

Simplify Each Term:

[Tex]\frac{(x – 1)^2}{x^2 – 2x + 1}[/Tex]

​Notice that x² – 2x + 1 = (x – 1)²

[Tex]=\frac{(x – 1)^2}{(x – 1)^2} [/Tex]

= 1

Simplify Second Term:

[Tex]\frac{(x + 1)^2}{x^2 + 2x + 1}[/Tex]

​Notice that x² + 2x + 1 = (x+1)²

[Tex]\frac{(x + 1)^2}{(x + 1)^2} = 1[/Tex]

Add simplified terms: 1 + 1 = 2

Hence [Tex]\frac{(x – 1)^2}{x^2 – 2x + 1} + \frac{(x + 1)^2}{x^2 + 2x + 1} = 2[/Tex]

Question 8: Simplify Algeabric Expression: [Tex]\frac{5x}{4} + \frac{3x}{4}[/Tex]

Solution:

= 5x/4 + 3x/4

Taking LCM of Denominator

= (5x + 3x)/4

= 8x/4 = 2x

Question 9: Simplify Algebraic Expression: 2x²+3x-4+x²-2x+1

Solution:

= 2x² + x²+ 3x – 2x – 4 + 1

= 3x² + x – 3

Question 10: Simplify Algebraic Expression: 2(x+3)-(x-4)+5(x+1)

Solution:

= 2(x+3) – (x-4) + 5(x+1)

= 2x + 6 – x + 4 + 5x + 5

= 6x + 15

Algebraic Expressions: Practice Problems

P 1: Solve the algebraic equation: [Tex]\frac{4x^2 – 9}{2x – 3} \cdot \frac{3x + 3}{x^2 – 1}[/Tex]

P 2: Solve the algebraic equation: [Tex] \frac{2x^2 – 8}{4x} + \frac{3x^2 + 6x}{2x}[/Tex]

P 3: Solve the algebraic equation: 7x+4x+2x+4y-2y+9x.

P 4: Solve the algebraic equation: [Tex]\frac{5x – 7}{x + 2} – \frac{3x + 1}{x + 2}[/Tex]

P 5: Solve the algebraic equation: -2(3x-6)

P 6: Solve the algebraic equation: [Tex]\frac{2x^3 – 6x^2 + 4x}{2x}[/Tex]

P 7: Solve the algebraic equation: [Tex]\frac{6x^2 – 5x – 6}{x^2 – 3x}[/Tex]

P 8: Solve the algebraic equation: [Tex]\frac{3x^2 – 5x + 2}{x^2 – x – 2}[/Tex]

P 9: Solve the algebraic equation: 9a-2a+74a – 6a

P 10: Solve the algebraic equation: [Tex]\frac{x^2 + 2x + 1}{x^2 – 1} + \frac{x^2 – 1}{x^2 + 2x + 1}[/Tex]

Also Check,

• How to Identify an Algebraic Expression?
• Are algebraic expressions polynomials?
• Practice questions on Rational numbers

Frequently Asked Questions

What is an Algebraic Expression?

Algebraic expression is a mathematical phrase that can include numbers, variables (like x or y), and operation symbols (like +, -, *, /). It represents a value and can be as simple as a single number or variable, or as complex as a combination of many terms.

What are Terms in an Algebraic Expression?

Terms are the individual parts of an algebraic expression separated by plus (+) or minus (-) signs. For example, in the expression 3x²+4x-5, there are three terms: 3x², 2x, and -4.

What are Coefficients and Constants in an Algebraic Expression?

• A coefficient is a numerical factor multiplied by a variable in a term. For example, in the term 2x, 2 is the coefficient.
• A constant is a term that does not contain any variables, such as 5 in the expression 2x + 5.

How to Simplify Algebraic Expressions?

To simplify an algebraic expression:

• Combine like terms (terms that have the same variable raised to the same power).
• Perform any indicated operations.

For example,

• Simplify 2x + 3x – 4
• Combine 2x and 3x to get 5x – 4

What is Difference between an Algebraic Expression and an Equation?

An algebraic expression is a combination of terms and operations that represent a value. An equation, on the other hand, states that two expressions are equal and includes an equal sign (=). For example, 2x + 3 is an expression, while 2x + 3 = 7 is an equation.