1.5 Linear Equations, Functions, and Applications

Linear Equations

Linear equations are foundational in algebra and are used to model a wide variety of real-world situations. Understanding how to solve and apply linear equations is essential for further study in mathematics and its applications.

Definition of a Linear Equation

• Linear Equation in One Variable: An equation that can be expressed in the form , where m and b are real numbers and .

Equivalent Equations

• Equivalent Equations: Equations that have the same solution set. For example, and are equivalent because $x = 1$ is the solution to both.

• Equations that do not have the same solution set are not equivalent. For example, and are not equivalent because $x^2 - 3x + 2 = 0$ has two solutions ($x = 1$ and ), but $x = 1$ only has one solution.

Equation-Solving Principles

• Addition Principle: If is true, then is also true for any real number .

• Multiplication Principle: If is true, then is also true for any real number .

Example: Solving a Linear Equation with Fractions

To solve , it is often convenient to multiply both sides by the least common denominator (LCD) to clear fractions.

Solution: Multiply both sides by 20 (the LCD of 4 and 5), then solve for :

The solution is .

Graphical Solution and the Intersect Method

• Graphical calculators can be used to solve equations by graphing both sides as separate functions and finding their intersection point.

• For , graph and ; the -coordinate of their intersection is the solution.

Solving Linear Equations: Algebraic and Graphical Methods

• Use algebraic manipulation (distributive property, combining like terms, addition/multiplication principles) to isolate the variable.

• Check the solution by substituting back into the original equation.

• Graphical solutions involve plotting both sides as functions and finding their intersection.

Special Cases in Linear Equations

• No Solution: If solving leads to a false statement (e.g., ), the equation has no solution.

• Infinitely Many Solutions: If solving leads to a true statement (e.g., ), any real number is a solution.

Applications Using Linear Models

Five Steps for Problem Solving

Solving real-world problems with linear equations involves a systematic approach:

1. Familiarize: Understand the problem, make drawings, list facts, assign variables, and organize information.

2. Translate: Express the problem in mathematical language or equations.

3. Carry Out: Use mathematical methods to solve the equation.

4. Check: Verify the solution in the context of the problem.

5. State: Clearly state the answer in a complete sentence.

Example: Studying Abroad

Given the total number of students and the difference between two groups, set up and solve a linear equation to find the number in each group.

The Motion Formula

The distance traveled by an object moving at rate for time is given by:

Example: Airplane Speed

To find when one plane overtakes another, set their distances equal and solve for time.

Example: Train Speeds

Similar to the airplane example, use the motion formula to determine when a faster train overtakes a slower one that had a head start.

The Simple-Interest Formula

The simple interest on a principal at interest rate for years is:

Example: Student Loans

Given the total amount borrowed and the total interest paid, set up a system of equations to find the amount borrowed at each rate.

Example: Investment Income

Given the total investment and total interest earned, use the simple interest formula to find how much was invested at each rate.

Applications: Geometry and Perimeter

Example: Solar Panels

Given relationships between length and width and the perimeter, set up and solve a linear equation to find the dimensions.

Summary Table: Key Linear Equation Applications