Study Notes on Systems of Linear Equations

Systems of Linear Equations in Two Variables

1. Solving Systems Using Substitution or Addition/Elimination Method

  • Objective: Solve the following systems of equations. If a system does not have one unique solution, classify it as inconsistent or dependent.
(a) Solve the following system:
  • Equations:
    • $-5(x + y) = 9 + 2y$
    • $6y - 2 = 10 - 7x$
  • Method: Choose either substitution or elimination.
  • Steps to solve:
    • Rearrange equations if necessary.
    • Use substitution or elimination to find $x$ and $y$.
  • Solution Check: Verify if there is one unique solution or if it is inconsistent.
(b) Solve the following system:
  • Equations:
    • $3x + y = 6$
    • $x + rac{1}{3}y = 2$
  • Method: Choose either substitution or elimination.
  • Steps to solve:
    • Convert $x + rac{1}{3}y = 2$ to a simpler form for substitution or elimination.
  • Solution Check: Determine the nature of the solution.
(c) Solve the following system:
  • Equations:
    • $3x - 4y = 9$
    • $2x + 9y = 2$
  • Method: Choose either substitution or elimination.
  • Steps to solve:
    • Rearrange both equations, if needed, for clear substitution or elimination.
  • Solution Check: Identify if the equations yield a unique solution or classification of the system.
(d) Solve the following system:
  • Equations:
    • $-4x - 8y = 2$
    • $2x = 8 - 4y$
  • Method: Choose either substitution or elimination.
  • Steps to solve:
    • Rearrange the second equation for clear substitution.
  • Solution Check: Confirm solution existence and classify.
(e) Solve the following system:
  • Equations:
    • $0.25x - 0.04y = 0.24$
    • $0.15x - 0.12y = 0.12$
  • Method: Choose either substitution or elimination.
  • Steps to solve:
    • Rewrite the equations if necessary to facilitate solving.
  • Solution Check: Establish if unique or dependent solutions exist.

2. Setting Up and Solving Linear Equations in Context

  • Task: Create and resolve systems of equations based on contextual scenarios. Clearly define all variables involved.
(a) Mixing Saline Solutions
  • Scenario: Walt wants to mix:
    • A 30% saline solution
    • A 10% saline solution
  • Goal: Create 200 mL of a 12% saline solution.
  • **Variables: **
    • Let $x$ = volume (in mL) of the 30% saline solution
    • Let $y$ = volume (in mL) of the 10% saline solution
  • Equations Setup:
    • $x + y = 200$
    • $0.30x + 0.10y = 0.12(200)$
  • Method: Solve the system using substitution or elimination to find values of $x$ and $y$.
(b) Fuel Consumption for a Sedan
  • Scenario: Chris drives:
    • 12 miles per gallon (mpg) in the city
    • 18 mpg on the highway
  • Total Distance: 420 miles
  • Total Fuel Used: 26 gallons
  • Variables:
    • Let $c$ = miles driven in the city
    • Let $h$ = miles driven on the highway
  • Equations Setup:
    • $c + h = 420$
    • $ rac{c}{12} + rac{h}{18} = 26$
  • Method: Solve the system to determine the miles driven in both city and highway.
(c) Speed of a Fishing Boat
  • Scenario: A boat travels:
    • 44 mi north with the current in 2 hours
    • 56 mi south against the current in 4 hours
  • Variables:
    • Let $b$ = speed of the boat in still water (in mph)
    • Let $c$ = speed of the current (in mph)
  • Equations Setup:
    • Speed with current: $b + c = rac{44}{2} = 22$
    • Speed against current: $b - c = rac{56}{4} = 14$
  • Method: Solve the system of equations to determine both the speed of the boat and the speed of the current.

3. Challenge Problem

  • Task: Solve for $x$ and $y$ given the following system:
  • Equations:
    • $\frac{1}{x} + 2y = 1$
    • $-\frac{1}{x} + 4y = -7$
  • Hint: Use substitutions:
    • Let $u = \frac{1}{x}$ and $v = \frac{1}{y}$ to rewrite the equations.
  • Solution Steps:
    • Substitute $u$ and $v$ into the equations and solve for $u$ and $v$ first, then revert back to $x$ and $y$.