Linear Equations and Zeros of Linear Functions

Linear Equations

Definition of Linear Equations in One Variable

A linear equation in one variable is an equation that can be expressed in the form ax + b = 0, where a and b are real numbers and a ≠ 0.

• Variable: The unknown quantity, usually represented by x.

• Linear: The variable is raised only to the first power (no exponents other than 1).

Equation-Solving Principles

To solve linear equations, two main principles are used:

• Addition Principle: For any real numbers a, b, c:

  • If a = b is true, then a + c = b + c is also true.

• Multiplication Principle: For any real numbers a, b, c (with c ≠ 0):

  • If a = b is true, then ac = bc is also true.

Solving Linear Equations with Fractions

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

• Example: Solve $\frac{3}{2}x - \frac{1}{5} = 7$

  • Multiply both sides by the LCD (which is 10) to eliminate denominators.

Special Cases in Linear Equations

Some linear equations may have no solution or infinitely many solutions, depending on how the variables and constants relate after simplification.

• Example: Solve $-24x + 7 = 17 - 24x$

  • After simplifying, if the variable terms cancel and a true statement remains (e.g., $7 = 17$ is false), there is no solution.

  • If a true statement remains (e.g., $0 = 0$), there are infinitely many solutions.

Zeros of Linear Functions

Definition and Properties

An input a of a function f is called a zero of the function if the output for the function is 0 when the input is a. That is, a is a zero of f if $f(a) = 0$.

• A linear function $f(x) = mx + b$ (with $m \neq 0$) has exactly one zero.

Finding the Zero of a Linear Function

• Set $f(x) = 0$ and solve for $x$.

• Example: Find the zero of $f(x) = 2x - 12$

  • Set $2x - 12 = 0$

  • Solve: $2x = 12 \implies x = 6$

Applications of Linear Equations

Word Problems and Real-World Contexts

• Distance Formula: The distance d traveled by an object moving at rate r in time t is given by $d = rt$.

• Mixture Problems: Problems involving combining solutions or mixtures of different concentrations.

• Student Loans, Interest, and Sales Commission: Linear equations can be used to model and solve for unknowns in financial contexts.

• Geometry Applications: Finding dimensions of geometric figures (e.g., the width of a rectangular field) using linear equations.

Example Applications

• Student Loan: If a student takes out a loan and repays it in equal monthly installments, the total amount paid can be modeled by a linear equation.

• Sales Commission: If a salesperson earns a base salary plus a commission per sale, the total earnings can be modeled as $\text{Total Earnings} = \text{Base Salary} + (\text{Commission Rate} \times \text{Number of Sales})$.

• Field Dimensions: If the width of a rectangular field is 12 ft less than its length and the perimeter is 94 ft, set up the equation $2l + 2(l - 12) = 94$ to solve for the length and width.

Summary Table: Key Properties of Linear Equations

Property	Description	Example
Standard Form	$ax + b = 0$	$3x - 5 = 7$
Zero of a Function	Value of $x$ where $f(x) = 0$	$f(x) = 2x - 12 \implies x = 6$
Distance Formula	$d = rt$	$d = 60 \times 2 = 120$ miles
Special Cases	No solution or infinitely many solutions	$2x + 3 = 2x + 5$ (no solution)