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2.5 – Linear Systems

Objectives

  • Calculate the solution to a system of linear equations.
  • Compute and graph the solutions to a system of linear inequalities.
  • Explore multiple techniques for determining the solution(s) to a system.

 

Key Terms

  • Coincide – To overlap, or in the case of lines, to be identical
    • {(x,y) : Ax + By = C}
    • It means these lines are the same line and, therefore, lie on top of each other!
    • The equation is the set of points in common
  • Feasible Region – A region on a graph that includes all possible solutions to a system of inequalities.
    • Linear programming can be broken down into seven steps:
      1. Define your variables
      2. List the given information
      3. Write linear inequalities (called constraints)
      4. Graph the linear inequalities to create a “feasibility region.”
      5. Find the corner points of the feasibility region
      6. Evaluate the corner points in an expression for the value to be optimized
      7. Answer the question
  • Ordered Pair – A combination of coordinates (x,y), that describe a point on a graph.
    • You can also write ordered pairs with any letters: (a,b), (p,q), etc.
    • There are no spaces after the comma
  • Parallel Lines – Two lines with equal slopes that never intersect.
  • System of Linear Equations – A set of two or more equations.
    • There are 3 options for the solution:
      • A single point, (x,y): ONE SOLUTION
      • Parallel lines that never intersect: NO SOLUTION
      • Two lines that coincide because they are the same line: INFINITE SOLUTIONS

Notes

Parallel & Coinciding Lines
Alg 2A 2.5 - Parallel Coincide
Intersecting Lines
  • Example
    • Jenn and Sal were solving a system of equations. They both noticed that the two lines had different slopes.
    • Jenn said that because each line in the system had different slopes, the two lines had to intersect, which meant there was one solution to the system.
    • Sal disagreed, and said they should also look at the y-intercepts. Who is correct?
      • Answer: Jenn is correct.
        • When two lines have different slopes, they must intersect, producing one solution.
        • Remember: only parallel lines never intersect. Parallel lines have the same slope. These two lines have different slopes, so they will eventually intersect.
        • The y-intercept has nothing to do with intersections.

 

Graphing Systems of Linear Equalities
Alg 2A 2.5 - Comparison Graphs
Graphing Systems of Linear Inequalities
  • Example
    • x + 2y \leq 4
    • 3x - y \textgreater 2

 

  • Step 1: Plug in x values to find y for each inequality or convert each inequality to slope-intercept form
    • Slope-intercept form of the 1st inequality
      • Subtract x from both sides, then divide by 2
      • Simplify and rearrange
      • y\leq\frac{-1}{2}x+2
      • Graph a solid line and shade (the purple area)
    • Slope-intercept form of the 2nd inequality
      • Subtract 3x from both sides, then divide by -1 (flip the sign)
      • Simplify and rearrange
      • y\textless 3x-2
      • Graph a dashed line and shade (the pink area)
  • Step 2: The overlap areas are the solution points (the green area shaded below)

Alg 2A 2.5 - System Inequality Graphs

 

Substitution Method of Solving a System of Equations
  • Example: If you were to solve the following system by substitution, what would be the best variable to solve for and from what equation?
    • 2x + 8y = 12
    • 3x – 8y = 11

 

  • Answer: x, in the first equation
    • Rationale (reason): the coefficients in the 1st equation are all reducible to give you x (without a coefficient). So, if you divide by 2 throughout all terms of the equation, you can isolate x more easily.
    • Solving:
      • Step 1: Write equation: 2x + 8y = 12
      • Step 2: Divide by 2 (all terms)
        • \frac{2x}{2}+\frac{8y}{2}=\frac{12}{2}, which results in: x + 4y = 6
      • Step 3: Subtract 4y on both sides
        • x + 4y – 4y = 6 – 4y, which is x = 6 – 4y
      • Step 4: Substitute the value for x into the 2nd equation
        • 3(6 – 4y) – 8y = 11
      • Step 5: Distribute 3 into the parenthesis
        • 18 – 12y – 8y = 11
      • Step 6: Combine like terms
        • 18 – 20y = 11
      • Step 7: Subtract 18 on both sides
        • -20y = 11 – 18
      • Step 8: Combine like terms
        • -20y = -7
      • Step 9: Solve by dividing by -20 on both sides
        • y=\frac{7}{20}
      • Step 10: Substitute the value of y into one of the equations to solve for x
        • 2x + 8(\frac{7}{20}) = 12
      • Step 11: Simplify
        • 2x + (\frac{14}{5}) = 12
      • Step 12: Subtract 14/5 on both sides
        • 2x = 12 - \frac{14}{5}
      • Step 13: Combine like terms by finding an equivalent fraction for 12 with the denominator of 5
        • 2x = \frac{60}{5} - \frac{14}{5}
        • 2x = \frac{46}{5}
      • Step 14: Divide both sides by 2
        • Remember: to divide by 2, you can multiply by the reciprocal 1/2
        • (\frac{1}{2})\bullet 2=\frac{46}{5}(\frac{1}{2})
        • x = \frac{46}{10}
      • Step 15: Simplify
        • x = \frac{23}{5}
      • Step 16: Write your answer as a coordinate pair (x,y)
        • (\frac{23}{5}, \frac{7}{20})
Elimination Method of Solving a System of Equations
  • Before you can Eliminate a variable term in 2 equations, you must figure out what to multiply or divide one (or two) equation(s) by to get either the x terms or y terms to be the same.
  • How: Find the value of Q in the following system so that the solution to the system is the line:  {(x,y) : x – 3y = 4}
    • The system of equations:
      • x – 3y = 4
      • Qz – 6y = 8
    • Ask yourself, “what do I multiply or divde from the 1st equation to get to the 2nd?”
      • Answer: 2
      • Rationale: If you multiply 2 throughout each term (on both sides) of the 1st equation, it will result in the 2nd equation.
  • In order to solve the following system of equations by subtraction, what could you do before subtracting the equations so that one variable will be eliminated?
    • 4x – 2y = 7
    • 3x – 3y = 15
    • Answer: Multiply the top equation by 3 and the bottom equation by 4.
    • Rationale (reason):
      • Multiply the top equation by 3 and the bottom equation by 4.
      • In order to subtract, you need coefficients to be equal.
      • You will get 12x in the top equation, and 12x in the bottom equation, which can then be subtracted.
    • Solving:
      • Step 1: Multiply 3 throughout the terms in the 1st equation (both sides)
        • 3(4x - 2y = 7) \rightarrow 12x - 6y = 21
      • Step 2: Multiply 4 throughout the terms in the 2nd equation (both sides)
        • 4(3x - 3y = 15) \rightarrow 12x - 12y = 60
      • Step 3: Subtract the 2nd equation from the 1st (straight down)
        • 6y = -39
      • Step 4: Divide both sides by 6
        • \frac{6y}{6} = \frac{-39}{6}
      • Step 5: Simplify (reduce)
        • y = \frac{-13}{2}
      • Step 6: Substitute and solve for x
        • Choose either of the two original equations
          • 4x - 2(\frac{-13}{2}) = 7
          • 4x + 13 = 7
      • Step 7: Subract 7 on both sides
        • 4x = -6
      • Step 8: Divide both sides by 4
        • \frac{4x}{4} = \frac{-6}{4}
      • Step 9: Simplify (reduce)
        • x = \frac{-3}{2}
      • Step 10: Write as an ordered pair (x,y)
        • (\frac{-3}{2},\frac{-13}{2})
Solution Methods – Review
Alg 2A 2.5 - Solution Methods

Examples

Feasible Region
  • P = 2x + 3y

Alg 2A 2.5 - Feasibility Region

  • Look for the 4 corners of the shaded region. Note each coordinate:
    (0,0)
    (0,8)
    (6,5)
    (8,0)

 

  • Test each point in the function (equation): P = 2x + 3y

2(0) + 3(0) = 0
2(0) + 3(8) = 24
2(6) + 3(5) = 27
2(8) + 3(0) = 16

  • The maximum value is 27
  • The minimum value is 0

 

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