Ch4: Systems of Linear Equations y x y x y x y = ½ x –3 Consistent y = (-2/3)x + 2 Independent Equations One Solution (x, y) y = ½ x - 3 y = ½ x + 2 Inconsistent No Solutions y = ½ x + 3 2y = x + 6 Consistent Dependent Equations Many solutions (x,y)

Solving Systems: Graphical Method Step 1: Graph both equations Step 2: Find the point of intersection y x x + 2y = 7 x – y = 4 x yx y 0 3 ½ Solution: (5, 1)

Solving Systems: Substitution Method y x x + 2y = 7 x - y = 4 Step 1: Solve for x or y in 1 equation Step 2: Substitute into the 2 nd equation Step 3: Solve algebraically to find 1 variable Step 4: Plug the ‘found variable’ back in and solve for the second variable. Step 1: x – y = 4 [add y to both sides] x = y + 4 Step 2: x + 2y = 7 (y + 4) + 2y = 7 Step 3: 3y + 4 = 7 3y = 3 y = 1 Step 4: x = y + 4 x = (1) + 4 x = 5 Solution (5, 1)

Solving Systems:Elimination/Addition Method x + y = 10 x – y = 8 2x = 18 2 x = 9 x + y = y = 10 y = 1 Solution (9, 1) x + 2y = 7 x - y = 4 2 x + 2y = 7 2x - 2y = 8 3x = x = 5 x – y = y = 4 y = 1 Solution (5, 1) 2A + 3B = -1 3A + 5B = A + 15B = -5 -9A - 15B = 6 A = 1 2A + 3B = -1 2(1) + 3B = B = -1 3B = -3 B = -1

Practice 1.Solving using the elimination/addition method 2x - y = 4 5x + y = 3 2.Solving using the substitution method x + 2y = 2 3x + 4y = 0 3. Solve using any way you choose 2x + 5y = 2 -3x + 4y = 20 Answers 1. (1, -2) 2. (-4, 3) 3.(-4, 2)

Applications Write a system of equations for each scenario and solve: 1 Bank Account Problem: Carlos has 2 bank accounts. He has seven times as much in his savings account as in his checking account. In all, he has $3,200 in the bank. Find out how much Carlos has in each account. 2. Coins in a Jar: There are 93 coins in a jar. The coins are quarters and dimes All together the coins total $ How many quarters and dimes are in the jar? 3. Dimensions of a Rectangle: A soccer field has a perimeter of 320 yards. The length measures 40 yards more than its width. What are the field dimensions? 4. Acid Mixture (Revisited): How many ounces of a 5% hydrochloric acid and 20% hydrochloric acid must be combined to get 10 oz of solution that is 12.5% acid?

© 2008 Pearson Prentice Hall. All rights reserved Section 4.3 Systems of Linear Equations and Problem Solving Suppose you mix an amount of 25% acid solution with an amount of 60% acid solution. You then calculate the acid strength of the resulting acid mixture. For which of the following results should you suspect an error in your calculations? a.) 14% b.) 32% c.) 55%

Event Ticket Problem Hilton University Drama Club sold 311 tickets for a play. Student tickets cost 50 cents each, nonstudent tickets cost $1.50. If total receipts were $385.50, find how many tickets of eqach type were sold.

Finding Break-Even A manufacturing company recently purchased $3000 worth of new equipment to offer new personalized stationery to its customers. The cost of producing a package of personalized stationery is $3.00 and it is sold for $5.50. Find the number of packages that must be sold for the company to break even. Let x = number of packages of personalized stationery C(x) = total cost for producing x packages of stationery R(x) = total revenue for selling x packages of stationery