Linear Equations, Functions, and Applications – Study Notes

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 $mx + b = 0$, where m and b are real numbers and $m \neq 0$.

Equivalent Equations

• Equivalent Equations: Equations that have the same solution set. For example, $2x + 3 = 5$ and $x = 1$ are equivalent because $x = 1$ is the solution to both.

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

Equation-Solving Principles

• Addition Principle: If $a = b$ is true, then $a + c = b + c$ is also true for any real number $c$.

• Multiplication Principle: If $a = b$ is true, then $ac = bc$ is also true for any real number $c$.

Example: Solving a Linear Equation with Fractions

To solve $\frac{3}{4}x - 1 = \frac{7}{5}$, 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 $x$:

The solution is $x = \frac{16}{5}$.

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 $\frac{3}{4}x - 1 = \frac{7}{5}$, graph $y_1 = \frac{3}{4}x - 1$ and $y_2 = \frac{7}{5}$; the $x$-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., $7 = 17$), the equation has no solution.

• Infinitely Many Solutions: If solving leads to a true statement (e.g., $3 = 3$), 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 $d$ traveled by an object moving at rate $r$ for time $t$ is given by:

$d = r \cdot t$

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 $I$ on a principal $P$ at interest rate $r$ for $t$ years is:

$I = Prt$

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

Application	Key Formula	Example
Motion	$d = r t$	Airplane/Train overtaking problems
Simple Interest	$I = Prt$	Student loans, investments
Geometry	Perimeter formulas	Finding dimensions of rectangles