2026-07-16

Content Standards

Cluster

Notation

Standard

C. Solve systems of equations.

M.A.REI.C.6 (F2Y)

Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.

M.A.REI.C.7 (F2Y)

Solve a system of two linear equations in two variables algebraically and graphically. (Systems involving a quadratic equation will be introduced later.)

Winter Trip

Imagine you are planning a winter trip to Door County or Wisconsin Dells. You need to rent a cabin. You find two different rental properties on Airbnb:

  • Cabin A (Dells): Charges a flat cleaning fee of $30, plus $10 per night.

  • Cabin B (Door County): Charges no upfront fee, but costs $20 per night.


Winter Trip

  • Cabin A (Dells): Charges a flat cleaning fee of $30, plus $10 per night.
  • Cabin B (Door County): Charges no upfront fee, but costs $20 per night.

1. If we plan to stay for 1 night which option is cheaper?

  • Cabin A: 1 night = $40
  • Cabin B: 1 night = $20

2. Suppose we want to figure out the number of nights in which both cabins cost the exact same amount?

We can solve this using a System of Linear Equations.

Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations sharing the same variables.

Depending on the equations, a linear system can have:

  • Exactly one solution: The lines intersect at one distinct point.
  • No solution: The lines are parallel and never intersect. This is an inconsistent system.
  • Infinitely many solutions: The equations describe the exact same line, so every point on that line works.

Systems of Equations, Define the Variables

Suppose we want to figure out the number of nights in which both cabins cost the exact same amount?

  • Cabin A (Dells): Charges a flat cleaning fee of $30, plus $10 per night.
  • Cabin B (Door County): Charges no upfront fee, but costs $20 per night.


Let \(x\) represent the number of nights, and let \(y\) represent the total cost in dollars.


  • Cabin A: \(y = 10x + 30\)
  • Cabin B: \(y = 20x\)

Systems of Equations, Solve using Substitution

At what point do both properties charge the exact same amount?

  • Cabin A: \(y = 10x + 30\)
  • Cabin B: \(y = 20x\)

Substitution Method:

\(10x + 30 = 20x\)

\(\Rightarrow\) \(30 = 20x - 10x\)

\(\Rightarrow\) \(30 = 10x\)

\(\Rightarrow\) \(\frac{30}{10} = x\)

\(\Rightarrow\) \(x = 3\)

Systems of Equations, Solving Graphically

https://www.desmos.com/calculator

Plot the two equations on a Cartesian coordinate plane in Desmos.

  • At what point do the lines intersect?

  • Use the graph to determine which of the properties is less expensive if we plan to stay 2 nights? 10 nights?