Example

Solving an Algebraic Equation

An algebraic equation is a mathematical statement that shows the equality of two expressions. The goal of solving an algebraic equation is to find the value(s) of the unknown variable(s) that make the equation true.

The general approach to solving an algebraic equation includes:

• Recognizing the type of equation (e.g., linear, quadratic, etc.).
• Isolating the variable on one side of the equation.
• Applying mathematical operations to simplify and solve for the variable.

Solving a Linear Equation

A linear equation is an equation where the highest power of the variable is 1. The general form of a linear equation is:

\[ ax + b = c \]

Example:

If the equation is \( 2x + 5 = 15 \), the solution is:

• Step 1: Subtract 5 from both sides: \( 2x = 10 \).
• Step 2: Divide both sides by 2: \( x = 5 \).

Solving a Quadratic Equation

A quadratic equation is an equation where the highest power of the variable is 2. The general form of a quadratic equation is:

\[ ax^2 + bx + c = 0 \]

Example:

If the equation is \( x^2 - 5x + 6 = 0 \), the solution is:

• Step 1: Factor the equation: \( (x - 2)(x - 3) = 0 \).
• Step 2: Set each factor equal to zero: \( x - 2 = 0 \) or \( x - 3 = 0 \).
• Step 3: Solve for \( x \): \( x = 2 \) or \( x = 3 \).

Solving an Equation with Fractions

If the equation includes fractions, you can solve it by multiplying both sides by the least common denominator (LCD) to eliminate the fractions.

Example:

If the equation is \( \frac{2}{3}x + \frac{1}{4} = 5 \), the solution is:

• Step 1: Multiply both sides by the LCD, which is 12: \( 12 \times \left( \frac{2}{3}x + \frac{1}{4} \right) = 12 \times 5 \).
• Step 2: Simplify: \( 8x + 3 = 60 \).
• Step 3: Subtract 3 from both sides: \( 8x = 57 \).
• Step 4: Divide both sides by 8: \( x = \frac{57}{8} \) or \( x = 7.125 \).

Real-life Applications of Solving Algebraic Equations

Solving algebraic equations has many practical applications, such as:

• Determining the amount of material needed to create something (e.g., determining the length of a fabric to make clothes).
• Solving problems related to budgeting and finance (e.g., determining how much money to save each month to reach a goal).
• Understanding patterns and relationships in scientific data (e.g., calculating the speed of an object in motion).

Common Operations with Algebraic Equations

Linear Equation: \( ax + b = c \)

Quadratic Equation: \( ax^2 + bx + c = 0 \)

Modifying Equations: If the equation involves fractions, powers, or roots, additional algebraic operations such as factoring, multiplying by a common denominator, or taking square roots may be required.