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:
  • $3x + y = 6$
  • $x + \frac{1}{3}y = 2$
• Method: Choose either substitution or elimination.
• Steps to solve:
  • Convert $x + \frac{1}{3}y = 2$ 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: $b + c = \frac{44}{2} = 22$
  • Speed against current: $b - c = \frac{56}{4} = 14$
• 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 $y$.