SYSTEMS OF EQUATIONS

Unit 2 Systems of Equations A system of equations is two or more equations with the same variables. ex. 3x-2y=8 4x+3y=12 There are several ways to solve systems. 1. solve by graphing 2. solve by substitution 3. solve by linear combination 4. solve using matrices

Solving by Graphing When solving a system by graphing you need to graph each line and look for a point of intersection. The point (x,y) is the solution to the system. To be a solution, it must satisfy both equations.

Classifying Systems If a system has at least one solution, then it is called consistent If a system does not have a solution then it is called inconsistent If the system is consistent and has one solution, then it is called independent If the system is consistent and has an infinite number of solutions, then it is called dependent.

Solving by Graphing If the lines intersect at one point, then there is one solution If the lines do not intersect at all, then there is no solution If the lines intersect at every point, then there is an infinite number of solutions

Examples

Solving Algebraically Substitution Method— 1.Solve one equation for one variable (best to solve for variable with a coefficient of 1 if possible) 2.Substitute for the variable you just solved for in first step into the other equation 3.Once you find the value of one variable, substitute into either equation to find value of other variable

Example

Unique Solutions When solving by substitution method and the variables cancel out and the statement you are left with is false, like 5=8, then there is no solution to the system When solving and the variables cancel out and the statement you are left with is true, then there is an infinite # of Solutions (not all reals!)

Examples

Linear Combination also called addition method or elimination method Add the two equations together so that one variable cancels out (may need to multiply one or both of the equations by some factor that would cause a variable to cancel Need to have equation in standard form

Examples

More examples

More Examples

Real World Examples Assign variables Set up equations Solve Ex. The perimeter of a rectangle is 36 inches. The length is twice the width. Find the width.

Real World Examples Ex: Thirty people are going to lunch. It costs $5 for kids and $12 for adults. The total bill without tax was $220. How many kids and adults went to lunch?

Systems with 3 equations and 3 variables Steps: 1.Take 2 equations and eliminate a variable 2.Take two different equations and eliminate the SAME variable 3.Take the 2 new 2 variable equations and eliminate one of those variables 4.Plug the value into either 2 variable eq. for the value of the 2 nd variable 5.Plug both values into original eq. to find value of 3 rd variable

Example

Work Space (7,4,-5)

More Examples

Work Space (4,-3,-1)

One more Example—don’t complain!!!

Work Space (2/3,2,-5)

Try one more…

(-2,3/5,2)

Homework

Hw answers

System of Linear Inequalities To graph a linear inequality, you graph the related linear equation and shade the area that contains the solution symbols have a boundary line that is dotted—if it is ≤ or ≥ then the boundary line is solid Shade the region that includes x and y values that make the inequality true (test a point)

System of Linear Inequalities, cont. 2 or more inequalities Graph each inequality and shade Darken where the two shaded regions overlap—this will be the solution to the system

Example

Linear Programming A method used to find optimal solutions such as maximum or minimum profits Steps: 1.Assign variables 2.Determine the constraints (inequalities) 3.Find the feasible region (area of solution) 4.Determine the vertices of feasible region 5.Plug those values into the profit equation(also called objective function)

Example Mr. Farmer wants to plant some corn and wheat and he gets the following statistics from the US Bureau of Census CropYield per acreAvg Price Corn113.5 bu$3.15/bu Soybeans34.9 bu$6.80 bu Wheat35.8 bu$4.45 bu Cotton540 lb$0.759/lb Rice, rough5621 lb$0.0865/lb

Example continued Mr. Farmer can have no more 120 acres of corn and wheat At least 20 and no more than 80 acres of corn At least 30 acres of wheat How many acres of each crop should Mr. Farmer plant to maximize the revenue from his harvest?

Working through the problem.. Assign Variables X=acres of corn and y=acres of wheat List the constraints

Graph it

Example continued List the vertices Determine Profit equation Which would yield the most?

Another Ex. A snack bar cooks and sells hamburgers and hot dogs during football games. To stay in business, it must sell at least 10 hamburgers but cannot cook more than 40. It must also sell at least 30 hot dogs but cannot cook more than 70. The snack bar cannot cook more than 90 items total. The profit on a hamburger is $0.33 and on a hot dog it is $0.21. How many of each item should it sell to make the maximum profit? Profit Equation: __________________________ Constraints: Answer: _________________________

Another Example As a receptionist for a veterinarian, Sue scheduled appointments. She allots 20 minutes for a routine office visit and 40 minutes for surgery. The vet can not do more than 6 surgeries per day. The office has 7 hours available for appointments. If an office visits costs $55 and most surgeries costs $125, find a combination of office visits and surgeries that will maximize the income the veterinarian practice receives per day.

What do you know… Assign variables x=number of office visits y=number of surgeries Constraints: 7 hours needs to be in terms of minutes

Continued… Graph and determine coordinates of the vertices You should get (0,0) (0,6)(9,6)(21,0)

Continued…. Determine the profit equation $55v + $125s = P Test the points Highest profit would be when there are 6 surgeries and 9 visits