Solving Systems of Equations Ch 7.1 Solving Systems of Equations

PRECALC RETAKE IF you get an A or B on the Ch 4/5 test, you may retake the quest AT LUNCH March 8 or 9th

Reviewing Linear Functions In-n-out activity Reviewing Linear Functions

In-n-Out ordinarily sells hamburgers, cheeseburgers, and Double-Doubles (two beef patties and two slices of cheese). While they don't advertise it, they have a secret menu which includes a burger where you can order as many extra beef patties and cheese slices you like. The most common orders are 3x3's (read as "three by three") and 4x4. However some people have ordered 20x20's and even a 100x100!

Draw on white board the two buns, cheese, and meat CHEESEBURGER

Draw on white board the two buns, two cheese, and two meat DOUBLE DOUBLE

20 X 20

100 X 100!!

How much does the 100 X 100 cost? Goal for today

How much does the 3 X 3 cost? Can someone explain or draw what the picture looks like? Use the table to help get you started. We will check that everyone has the same price for 3X3

$90.85 $97.66 with 7.5% sales tax

How much does an n X n cost?

What if INNOUT invests 10,000 in equipment to produce a new milkshake What if INNOUT invests 10,000 in equipment to produce a new milkshake. Each milkshake costs $0.65 to produce, but it will be sold for $1.20. How many milkshakes would innout need to sell in order to break even? This is the kind of question that business owners might ask. What do they need to solve this problem?

What is a “break even” point and how do we solve for it? When we graph a Cost equation and a Revenue equation we can see very easily where the Break-even point is. After this point the company will be making a gain and before this point they will be suffering a Loss. The extent of that gain or loss is seen in the graph. What is a “break even” point and how do we solve for it? R(x) = c(x)

EX 2 A small business invests $6,000 in equipment to produce a new soft drink. The unit cost to produce is $23.20, and the selling price is $35.20. How many unit must be sold in order to break even?

Ch 7.1 Notes A systems of equations is two or more equations in two or more variables. A solution of this system is an ordered pair that satisfies each equation in the system. 2x + y = 5 3x – 2y = 4 Which ordered pair satisfies both equations? (4,3) (2,1) or (5,6)

Method of Graphing 2x + y = 5 3x – 2y = 4 When you use this method, the solution is the point of intersection because that is where the (x,y) values are equal. To use this method, Graph each equation ( it might be Helpful to solve for y in terms of x 2) Find the intersection point (on a Calculator, use the intersect feature. See next slide for more details) 2x + y = 5 3x – 2y = 4

You try! Graph the system. Find the intersection points. Then solve the system algebraically (hint: substitution might be easiest). SHOW ALL WORK. -2x + y = - 5 x² + y² = 25

Classifying Answers when solving Equations “No solution”

Method of Substitution To use this method, 1) Solve one of the equations in terms of the other 2) Substitute the expression found in step 1 into the other equation to obtain an equation in one variable 3) Solve the equation obtained in Step 2. 4) Back substitute the values obtained in Step 3 into the expression obtained in step 1 to find the values of the other variable 5) Check that each solution satisfies both of the original equations Solve the Original Example with this method! 2x + y = 5 3x – 2y = 4

EXAMPLE Solve the system of equations x ² + 4x – y = 7 2x – y = -1 Solutions ( -4, -7) and (2, 5)

Check by Graphing Solutions of x ² + 4x – y = 7 2x – y = -1 (4,-7) and ( 2,5)

EX With no Solution -x + y = 4 x² + y = 3

Method of Elimination Solve the Original Example with this method! Obtain coefficients for x (or y) that differ only in sign by multiplying all terms of one of both equations by suitable chosen constants Add the equations together. Back substitute the value obtained in step 2 into either of the original equations and solve for the other variable Check your solution in both of the original equations Solve the Original Example with this method! 2x + y = 5 3x – 2y = 4 Solve the Original Example with this method!

EXAMPLE 5 Solve the system of equations y = ln(x) x + y = 1 Which of the methods would be appropriate here? (1, 0) Graphing!

PRACTICE WS SIDE #1

EX 7

EX 6: Profit and Cost (ex 2 pg 476) A total of $12,000 is invested in two funds paying 9% and 11% simple interest. The yearly interest is $1180. How much is invested at each rate? X = 7000 Y = 5000 X = 7000 Y = 5000

HW Pg 481 #5-9, 23, 25