Basic Algebraic Operations

Basic Algebraic Operations

Understanding Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. A variable is a symbol that represents an unknown value, often denoted by letters such as x or y. A number is a constant value. The operations include addition (+), subtraction (−), multiplication (×), and division (÷).

For example, the expression 3x + 5 represents a combination of the variable x, the constant 3, and the constant 5. Here, 3 is multiplied by x, and then 5 is added.

Order of Operations

When evaluating algebraic expressions, it is important to follow the order of operations. The order is as follows:

1. Brackets
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Example of Order of Operations

Consider the expression 2 + 3 × (4 − 1)^2. To evaluate this:

1. Start with the brackets: 4 − 1 = 3.
2. Now the expression is 2 + 3 × 3^2.
3. Calculate the exponent: 3^2 = 9.
4. The expression is now 2 + 3 × 9.
5. Next, perform multiplication: 3 × 9 = 27.
6. Finally, add: 2 + 27 = 29.

The result is 29.

Combining Like Terms

Like terms are terms in an expression that have the same variable raised to the same power. To combine like terms, you simply add or subtract their coefficients.

For example, in the expression 4x + 3x − 2y + 5y, the like terms are:

• 4x and 3x
• −2y and 5y

To combine these:

1. 4x + 3x = 7x
2. −2y + 5y = 3y

The simplified expression is 7x + 3y.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This means you can distribute a factor across a sum or difference.

Example of the Distributive Property

Consider the expression 3(x + 4). To apply the distributive property:

1. Multiply 3 by x: 3x.
2. Multiply 3 by 4: 12.

The result is 3x + 12.

Solving Linear Equations

A linear equation is an equation of the first degree, which means it has no exponents greater than one. The general form is ax + b = c, where a, b, and c are constants.

Example of Solving a Linear Equation

To solve the equation 2x + 3 = 11:

1. Subtract 3 from both sides: 2x + 3 − 3 = 11 − 3.
2. This simplifies to 2x = 8.
3. Now divide both sides by 2: x = 8 ÷ 2.
4. The solution is x = 4.

Working with Fractions

Fractions are expressions that represent a part of a whole. The top number is the numerator, and the bottom number is the denominator. To perform operations with fractions, you need to find a common denominator.

Example of Adding Fractions

To add the fractions 1/4 and 1/6:

1. Find the least common denominator (LCD). The LCD of 4 and 6 is 12.
2. Convert the fractions: 1/4 = 3/12 and 1/6 = 2/12.
3. Add the fractions: 3/12 + 2/12 = 5/12.

The result is 5/12.

Exponents and Their Properties

An exponent indicates how many times a number (the base) is multiplied by itself. For example, in 2^3, 2 is the base, and 3 is the exponent, which means 2 × 2 × 2 = 8.

Properties of Exponents

• Product of Powers: a^m × a^n = a^(m+n)
• Quotient of Powers: a^m ÷ a^n = a^(m−n)
• Power of a Power: (a^m)^n = a^(m×n)

Example of Using Exponent Properties

Consider the expression 2^3 × 2^2:

1. Using the product of powers: 2^(3+2) = 2^5.
2. This simplifies to 32.

Introduction to Polynomials

A polynomial is an algebraic expression that consists of terms, where each term includes a coefficient and a variable raised to a non-negative integer exponent. The general form is a_nx^n + a_(n−1)x^(n−1) + ... + a_1x + a_0.

For example, 2x^2 + 3x + 5 is a polynomial with three terms.

Summary

• An algebraic expression consists of numbers, variables, and operations.
• Follow the order of operations: PEMDAS.
• Combine like terms by adding or subtracting their coefficients.
• Use the distributive property to simplify expressions.
• A linear equation can be solved by isolating the variable.
• Fractions require a common denominator for addition or subtraction.
• Exponents indicate repeated multiplication and follow specific properties.

Check your understanding

• What is the result of the expression 5 + 2 × (3 + 1)?
• How do you combine the like terms 6x + 2x − 4y + 3y?
• What is the solution to the equation 3x − 7 = 2?
• How do you add the fractions 1/3 and 1/4?