# 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
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
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)

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?”
• 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

## Examples

Feasible Region
• P = 2x + 3y

• 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