Applications of Systems of Linear Equations

Solving Applied Problems Using a System of Equations

Systems of linear equations are powerful tools for solving real-world problems involving multiple unknowns. The process involves translating a word problem into mathematical equations and solving for the variables.

• Step 1: Understand the Problem – Identify what information is given and what needs to be found.

• Step 2: Assign Variables – Let variables represent the unknown quantities.

• Step 3: Write a System of Equations – Use the information to write two or more equations involving the variables.

• Step 4: Solve the System – Use algebraic methods such as substitution or elimination to find the values of the variables.

Example: Soccer Field Dimensions

Problem: A soccer field must have a width between 50–100 yards and a length between 100–130 yards. The field has a perimeter of 320 yards, and its length is 40 yards more than its width. Find the dimensions.

• Assign Variables: Let = length, = width.

• Write Equations:

  • Perimeter formula:

  • Length in terms of width:

• Solve the System (by substitution):

  • Substitute in the perimeter equation:

  • Expand:

  • Combine like terms:

  • Subtract 80:

  • Divide by 4:

  • Find :

• Solution: The width is 60 yards, and the length is 100 yards.

Additional info: The solution checks that the dimensions fit within the required ranges and satisfy the perimeter condition.

Solving Money Problems Using Two Variables

Systems of equations can also be used to solve problems involving the cost of multiple items when the total cost is known for different combinations.

• Step 1: Assign Variables – Let each variable represent the unknown price of an item.

• Step 2: Write the System – Use the given total costs to write equations.

• Step 3: Solve the System – Use substitution or elimination to find the values.

Example: Ticket Prices

Problem: Two football tickets and one basketball ticket cost $283.42. One football ticket and two basketball tickets cost $219.95. What are the average ticket prices for football and basketball?

• Assign Variables: = average football ticket price, = average basketball ticket price.

• Write Equations:

• Solve the System (by elimination):

  • Multiply the second equation by 2:

  • Subtract the first equation:

  • Substitute into the first equation:

• Solution: The average football ticket price is $115.63, and the average basketball ticket price is $52.16.

Verification:

• Check in both equations:

Summary Table: Steps for Solving Word Problems with Systems of Equations

Additional info: These methods are foundational in College Algebra and are widely applicable in fields such as business, science, and engineering.