Solving Linear Equations and Applications in College Algebra

Solving Linear Equations with One Variable

Definition of a Linear Equation

A linear equation in one variable is an equation that can be written in the form:

• $ax + b = 0$

• where a and b are real numbers, and a ≠ 0.

Generating Equivalent Equations: An equation can be transformed into an equivalent equation by performing the same operation on both sides.

Steps to Solve Linear Equations

1. Simplify each side of the equation (combine like terms, distribute).

2. Add or subtract the same real number from both sides to isolate terms with the variable.

3. Multiply or divide both sides by the same nonzero real number to solve for the variable.

4. Check the solution by substituting it back into the original equation.

Example:

• Solve $3x - 6 = 15$

• Add 6 to both sides: $3x = 21$

• Divide both sides by 3: $x = 7$

• Check: $3(7) - 6 = 21 - 6 = 15$

Solving Linear Equations Involving Fractions

Clearing Fractions

When an equation contains fractions, it is often helpful to clear the denominators by multiplying both sides by the least common denominator (LCD).

• Identify the LCD of all denominators.

• Multiply both sides of the equation by the LCD to eliminate fractions.

• Solve the resulting linear equation.

Example:

• Solve $\frac{x}{3} - \frac{5}{6} = \frac{x}{2}$

• LCD is 6. Multiply both sides by 6:

• $6 \left(\frac{x}{3}\right) - 6 \left(\frac{5}{6}\right) = 6 \left(\frac{x}{2}\right)$

• $2x - 5 = 3x$

• Subtract $2x$ from both sides: $-5 = x$

• Check: $\frac{-5}{3} - \frac{5}{6} = \frac{-5}{2}$

Solving Rational Equations with Variables in the Denominator

Rational Equations

A rational equation is an equation involving one or more rational expressions (fractions with variables in the denominator).

• Find the least common denominator (LCD) of all rational expressions.

• Multiply both sides by the LCD to clear denominators.

• Solve the resulting equation.

• Check for extraneous solutions (values that make any denominator zero are not valid solutions).

Example:

• Solve $\frac{5}{2x} = \frac{15}{x} - \frac{1}{3x}$

• LCD is $6x$. Multiply both sides by $6x$:

• $6x \cdot \frac{5}{2x} = 6x \cdot \frac{15}{x} - 6x \cdot \frac{1}{3x}$

• $15 = 90 - 2$

• Solve for $x$ and check for extraneous solutions.

Identities, Conditional Equations, and Inconsistent Equations

Types of Equations

• Identity: An equation that is true for all real numbers (e.g., $x + 2 = x + 2$).

• Conditional Equation: An equation that is true for at least one real number (e.g., $2x = 4$ is true when $x = 2$).

• Inconsistent Equation: An equation that is never true (e.g., $x + 2 = x + 3$).

Example:

• Solve $7x + 9 = 9(x + 1) - 2x$

• Simplify: $7x + 9 = 9x + 9 - 2x$

• $7x + 9 = 7x + 9$

• This is an identity (true for all real numbers).

Solving Applied Problems Using Mathematical Models

Modeling with Linear Equations

Many real-world problems can be modeled using linear equations. The process involves translating a word problem into an equation, solving for the unknown, and interpreting the result in context.

• Identify variables and write an equation based on the problem statement.

• Solve the equation for the unknown variable.

• Interpret the solution in the context of the problem.

Example:

• If the low-humor group has an average level of depression of $D = 10 - 0.8x$ and the high-humor group has $D = 9 - 0.6x$, where $x$ is the number of sessions, find $x$ when $D = 5.2$ for the low-humor group.

• $5.2 = 10 - 0.8x$

• $0.8x = 10 - 5.2 = 4.8$

• $x = 6$

Section 1.2 Homework Practice

Sample Problems

• Solve $8x - 12 = 60$

• Solve $\frac{2}{x} + 2 = \frac{12}{x+2}$

• Solve $\frac{1}{x+1} + 4 = \frac{17}{x-1}$

These problems reinforce the techniques of solving linear equations, clearing fractions, and checking for extraneous solutions.

Summary Table: Types of Equations

Type	Description	Example
Identity	True for all real numbers	$x + 2 = x + 2$
Conditional	True for at least one real number	$2x = 4$
Inconsistent	Never true	$x + 2 = x + 3$

Additional info: The notes include both worked examples and practice problems, providing a comprehensive overview of solving linear equations, including those with fractions and rational expressions, as well as classifying equations by their solution sets.