Print this Page
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:
 Define your variables
 List the given information
 Write linear inequalities (called constraints)
 Graph the linear inequalities to create a “feasibility region.”
 Find the corner points of the feasibility region
 Evaluate the corner points in an expression for the value to be optimized
 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 yintercepts. 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 yintercept has nothing to do with intersections.

Graphing Systems of Linear Equalities 

Graphing Systems of Linear Inequalities 
 Step 1: Plug in x values to find y for each inequality or convert each inequality to slopeintercept form
 Slopeintercept form of the 1st inequality
 Subtract x from both sides, then divide by 2
 Simplify and rearrange
 Graph a solid line and shade (the purple area)
 Slopeintercept form of the 2nd inequality
 Subtract 3x from both sides, then divide by 1 (flip the sign)
 Simplify and rearrange
 Graph a dashed line and shade (the pink area)
 Step 2: The overlap areas are the solution points (the green area shaded below)

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:
 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)
 Step 2: Multiply 4 throughout the terms in the 2nd equation (both sides)
 Step 3: Subtract the 2nd equation from the 1st (straight down)
 Step 4: Divide both sides by 6
 Step 5: Simplify (reduce)
 Step 6: Substitute and solve for x
 Choose either of the two original equations
 Step 7: Subract 7 on both sides
 Step 8: Divide both sides by 4
 Step 9: Simplify (reduce)
 Step 10: Write as an ordered pair (x,y)

Solution Methods – Review 

Examples
Feasible 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

Important!
Practice (Apex 2.5)
 Practice: Pgs 6, 8, 11, 14, 17, 18
 Watch the animations on Pgs 7, 9, 13
Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=1268