Systems

Day 1

Systems of Linear Equations System of Linear Equations: two or more linear equations together The solution of the system of equations the point of intersection of the two lines (makes both equations true) =

Example: Solve by Graphing a.) Step 1: Graph both equations on the same coordinate plane Step 2: Find point of intersection (2,1) Step 3: Check Answer The solution is (2,1)

Example: Solve by graphing b.) Step 1: Graph both equations on the same coordinate plane Step 2: Find point of intersection (-1,4) Step 3: Check Answer The solution is (-1,4)

Infinite Solutions & No Solutions A system of linear equations has NO SOLUTION when the graphs of the equations are parallel (same slope & different y-intercept). A system of linear equations has an INFINITE NUMBER OF SOLUTIONS when the graphs of the equations are the same line (same slope & y-intercept).

Example: Solve by graphing a.) have to put in slope-intercept form Equations have same slopes & different y-intercepts; therefore they are parallel Answer: No Solution

Example: Solve by graphing b.) have to put in slope-intercept form Equations have the same slope & same y-intercept; therefore, they are the same line Answer: Infinite # of Solutions

Day 2

Example: Solve by Graphing a.) Step 1: Graph both equations on the same coordinate plane Step 2: Find point of intersection (1,5) Step 3: Check Answer The solution is (1,5)

Example: Solve by Graphing b.)have to put in slope-intercept form Equations have same slopes & different y-intercepts; therefore they are parallel Answer: No Solution

Example: Solve by Graphing c.)have to put in slope-intercept form Equations have same slopes & same y-intercepts; therefore they are the same line Answer: Infinite # Solutions

Day 1

Example: Solve using Substitution a.) Step 1: Get one of the variables by itself on one side of the equation Step 2: Plug into the OTHER equation & solve for variable Step 3: Plug answer back into EITHER of the original equations to get 2 nd variable Solution:

Example: b.) Step 1: Get one of the variables by itself on one side of the equation Step 2: Plug into the OTHER equation & solve for variable Step 3: Plug answer back into EITHER of the original equations to get 2 nd variable Solution:

Example: c.) Need to get a variable by itself Now, our equations are: or Solution:

Example: d.) Need to get a variable by itself (doesn’t always have to be y) Now, our equations are: Solution:

Homework Worksheet: Solve by Substitution #1-8

Day 2

Example: Solve using substitution a.) Variables cancelled out. Left with a true statement? TRUE! Answer: Variables cancelled out. Left with a true statement? NOT TRUE! Answer: NO Solution b.) Infinite # of Solutions

Homework Worksheet: Solve by Substitution #9-21

Example: Your school committee is planning an field trip for 193 students. There are eight drivers available and two types of vehicles, school buses and minivans. The school buses seat 51 people each and the minivans seat 8 people each. How many buses and minivans will be needed?

Example: You have 11 bills in your wallet, some are $5 bills and some are $10 bills. You have a total of $95 in your wallet. How many $5 bills and how many $10 bills do you have?

Day 1

Example: Solve using Elimination a.) Step 1: Get one pair of variables that will cancel -6y and 6y will cancel Step 2: Add equations & solve for remaining variable Step 3: Plug 1 st variable into EITHER equation to get 2 nd variable Solution:

Example: b.) Step 1: Multiply one equation by a # to get a pair of variables that will cancel Step 2: Add equations & solve for remaining variable Step 3: Plug 1 st variable into EITHER equation to get 2 nd variable Solution:

Example: c.) Have to multiply BOTH equations to get a variable to cancel Can plug back into ANY of the equations Solution:

Homework: Worksheet: Solve by Elimination #1-8

Day 2

Example: Solving using elimination a.)b.)

Homework Worksheet: Solving by Elimination #9-21

Solving by Substitution

Systems with Three Variables The graph of any equation in the form Ax + By + Cz = D is a plane. The solution of a three-variable system is the intersection of the three planes.

When the solution of a system of equations in 3 variables is represented by one point, you can write it as an ordered triple: (x, y, z) (alphabetical order)

Example: Solve using substitution a.) Step 1: Choose one equation to solve for one of its variables Step 2: Substitute the expression into each of the other equations. Step 3: Write the two new equations as a system. Solve for both variables. Step 4: Plug those variables to one of the original equations to get remaining variable.

Example: Solve using substitution b.) Step 1: Choose one equation to solve for one of its variables Step 2: Substitute the expression into each of the other equations. Step 3: Write the two new equations as a system. Solve for both variables. Step 4: Plug those variables to one of the original equations to get remaining variable.

Solving by Elimination

Example: Solve using elimination a.) Step 1: Pair the equations to eliminate y, since the y- terms are already additive inverses. Add the equations. Step 2: Write the two new equations as a system, solve for the other two variables. Step 3: Substitute values in one of the original equations to solve for last remaining variable.

Example: Solve using elimination b.) Step 2: Pair the equations to eliminate y, since the y- terms are already additive inverses. Add the equations. Step 3: Write the two new equations as a system, solve for the other two variables. Step 4: Substitute values in one of the original equations to solve for last remaining variable. Step 1: Find the LCM for the coefficients of the variable you want to cancel & multiply the equations.

System of Linear Inequalities Shading: Plug in (0,0) NOT True! Shade on side without (0,0) NOT True! Shade on side without (0,0) The solution is where the shading overlaps Solution

Example: Solve by Graphing a.) use slope-int. form use x & y intercepts x=4y=3 Shading: Plug in (0,0) True! Solution!!!

Example: b.) x=2y=1 Shading: Plug in (0,0) NOT True! True!

Homework Worksheet: Solving Systems of Inequalities w/ 2 Equations

Word Problems

Example: A zoo keeper wants to fence a rectangular habitat for goats. The length should be at least 80ft & the distance around it should be no more than 310 ft. What are possible dimensions? x = width of habitat y = length of habitat x=155 y= Shading: Plug in (0,0) NOT True! True! Length Width Possible dimensions: Means: Width is 20 & Length is 100

Example: Suppose you want to fence in a rectangular garden. The length needs to be at least 50 ft & the perimeter to be no more than 140 ft. Solve by graphing. x = width of garden y = length of garden x=70 y=70 Shading: Plug in (0,0) Length Width NOT True! True!

Example: Solve by Graphing a.)

Example: Solve by Graphing b.)

Recall: +h moves LEFT -h moves RIGHT +k moves UP -k moves DOWN + V faces up - V faces down

Example: Solve by Graphing c.)

Example: Solve by Graphing d.)

Day 1

Linear Programming: a technique that identifies the minimum or maximum value of some quantity. This quantity is modeled with an objective function. Limits on the variables in the objective function are constraints, written as linear inequalities.

Example: Suppose you want to buy some tapes & CDs. You can afford as many as 10 tapes and 7 CDs. You want at least 4 CDs & at least 10 hours of recorded music. Each tape holds about 45 minutes of music and each CD holds about an hour. a.) Write a system of inequalities. x = #tapes purchased y = #CDs purchased as many as 10 tapes as many as 7 Cds at least 4 CDs at least 10 hours These inequalities model the constraints on x & y.

b.) Graph the system of inequalities

The shaded region in the graph is the feasible region & it contains all the points that satisfy all the constraints.

Say you buy tapes at $8 each & CDs at $12 each. The objective function for total cost C is If your total cost is $140, the equation would be 140 = 8x +12y, shown by the yellow line If your total cost is $112, the equation would be 112 = 8x +12y, shown by the purple line

As you can see, graphs of the objective function for various values of C are parallel. Lines closer to the origin (0, 0) represent lower costs. The graph closest to the origin that intersects the feasible region intersects it at the vertex (8, 4). The graph farthest from the origin that intersects the feasible region intersects it at the vertex (10, 7).

Graphs of an objective function that represent a maximum or minimum value intersect a feasible region at a vertex. Vertex Principle of Linear Programming If there is a maximum or a minimum value of the linear objective function, it occurs at one or more vertices of the feasible region.

Example: Find the values of x & y that maximize and minimize P for the objective function What is the value of P at each vertex? Constraints Step 1: Graph the Constraints

Example: Find the values of x & y that maximize and minimize P for the objective function What is the value of P at each vertex? Step 2: Find coordinates for each vertex Vertex

Example: Find the values of x & y that maximize and minimize P for the objective function What is the value of P at each vertex? Step 3: Evaluate P at each vertex Vertex When x = 4 and y = 3, P has its maximum value of 18. When x = 0 and y = 0, P has its minimum value of 0.

Homework: Worksheet: Textbook page 138 #1-3 Graph is already done for you, just have to do steps #2 & 3 (all work on separate paper)

Word Problem

Example: Suppose you are selling cases of mixed nuts and roasted peanuts. You can order no more than 500 cans and packages & spend no more than $600. How can you maximize your profit? How much is the maximum profit?

Continued…. Define variables Write Constraints x = # of cases of mix nuts ordered y = # of cases of roasted peanuts ordered no more than a total of 500 cans/packages spend no more than $600 can we have negatives?

Continued… Write objective function We need to write an equation for the profit, since that is what we are trying to maximize.

Continued… Graph constraints reduce!

Continued… Find & test vertices You can maximize the profit by selling 15 cases of mixed nuts & 16 cases of roasted peanuts. The maximum profit is $510.

Homework Worksheet 3-4 Use graph paper!