Solving Linear Equations and Applications

Outcomes When Solving Linear Equations

When solving a linear equation in one variable, there are three possible outcomes:

• One Solution (Conditional Equation): The equation is true for exactly one value of the variable.

• No Solution (Contradiction/Inconsistent): There is no value that makes the equation true. The solution set is the empty set.

• Infinitely Many Solutions (Identity): The equation is true for all real numbers; every value of the variable satisfies the equation.

Solving Formulas for a Specified Variable

It is often necessary to rearrange a formula to solve for a particular variable. This process is called solving for a variable or literal equations.

• Example: The distance formula is . To solve for time :

Practice Problems

• 1. Solve for :

  • Subtract from both sides:

  • Divide both sides by 2:

• 2. Solve for :

  • Multiply both sides by 2:

  • Divide both sides by :

• 3. Solve for :

  • Multiply both sides by 2:

  • Divide both sides by :

• 4. Solve for :

  • Expand right side:

  • Add to both sides:

  • Divide both sides by 3:

• 5. Solve for :

  • Multiply both sides by 3:

  • Divide both sides by :

Applications of Linear Equations

Linear equations are used to solve a variety of real-world problems, including percentage, interest, money, and geometry problems.

Percentage Problems

• Example 1: After a 30% reduction, a television costs $336. What was the original price?

Let be the original price. After a 30% reduction, the price is 70% of the original:

• Example 2: Including a 10.5% hotel tax, a room costs $216.58 per night. What is the cost before tax?

Let be the pre-tax cost. The total cost is $x$ plus 10.5% of $x$:

Interest Problems

• Example 3: A sum is split into two accounts: one at 2% interest, the other (which is $3000 more) at 3%. The total annual interest is $690. How much was invested at each rate?

Let be the amount at 2%, so is at 3%. Amount at 2%: $12,000; Amount at 3%: $15,000

Money Problems

• Example 4: A repair shop charges $448: $63 for parts, the rest for labor at $35/hour. How many hours of labor?

Labor cost:

• Example 5: Comparing two streaming services:

Let be the number of movies for which the total cost is the same: Total cost:

Geometry Problems

• Example 6: The perimeter of a rectangular field is 200 ft, and the width is 20 ft more than the length. Find the dimensions.

Let = length, . Perimeter: ,

• Example 7: A rectangular pool is three times as long as it is wide. Perimeter is 320 ft. Find dimensions.

Let = width, . Perimeter: ,

Summary Table: Types of Linear Equation Solutions

Key Reminders

• Homework and quiz due dates are important for success.

• Attendance and participation may affect your grade.

• Utilize office hours for additional help.