Warm-up Using n as the variable, write an equation. 1. 7 more than a number is 55. 2. 16 is 5 less than a number. 3. 6 less than twice Sarah s age is 20. 4. 47 is 11 more than three times Neil s age. 5. Dave earned $56.30. This was $4.50 less than Ned s earnings. What did Ned earn?
Warm-up Using n as the variable, write an equation. 1. 7 more than a number is 55. n + 7 = 55 2. 16 is 5 less than a number. 16 = n 5 3. 6 less than twice Sarah s age is 20. 2n 6 = 20 4. 47 is 11 more than three times Neil s age. 47 = 11 + 3n 5. Dave earned $56.30. This was $4.50 less than Ned s earnings. What did Ned earn? 56.30 = n 4.50
Homework Questions?
Section 9.3 Solving Problems with Two Variables
Objective I want to be able to use systems of linear equations in two variables to solve problems.
Example 1 The sum of two numbers is 5. The larger number exceeds twice the smaller number by 14. Find the numbers. Let n = larger number and k = smaller number n + k = 5 and n 2k = 14
Example 1 (continued) Since n + k = 5 and n = 2k + 14 (2k + 14) + k = 5 3k = -9 k = -3 n = 2(-3) + 14 = 8 The numbers are 8 and -3.
Example 2 A local shelter had 23 puppies and kittens adopted one week. There were 9 more puppy adoptions than kittens. How many of each were adopted? Let p = # of puppies; k = # of kittens p + k = 23 p = k + 9
There were 16 puppies and 7 kittens that were adopted. Example 2 (continued) Since p + k = 23 and p = k + 9 (k + 9) + k = 23 2k = 14 k = 7 p = 7 + 9 = 16
Example 3 A bank teller has 112 $5-bills and $10- bills for a total of $720. How many of each does the teller have? Let f = # of $5-bills and t = # of $10-bills f + t = 112 5f + 10t = 720
The teller has 80 $5-bills and 32 $10- bills. Example 3 (continued) Since f + t = 112 and 5f + 10t = 720 5f + 10(112 f) = 720 5f + 1120 10f = 720 400 = 5f f = 80 t = 112 80 = 32
Section 9.4 The Addition-or-Subtraction Method
Objective I want to be able to use addition or subtraction to solve systems of linear equations in two variables.
The Addition-or-Subtraction Method To solve a system of linear equations in two variables: 1. Add or subtract the equations to eliminate one variable. 2. Solve the resulting equation for the other variable. 3. Substitute in either original equation to find the value of the first variable. 4. Check in both original equations.
Example 1 Solve by the addition-or-subtraction method. x + y = 14 x y = 4 x + y = 14 x y = 4 2x + 0= 18 2x = 18 x = 9
Example 1 continued Solve by the addition-or-subtraction method. x + y = 14 x y = 4 x = 9 x + y = 14 (9) + y = 14 y = 5 The solution is (9, 5).
Example 1 continued Solve by the addition-or-subtraction method. x + y = 14 x y = 4 Check (9, 5) x + y = 14 x y = 4 9 + 5 = 14 9 5 = 4 14 = 14 4 = 4
Example 2 Solve by the addition-or-subtraction method. 3r + 2s = 2 3r + s = 7 3r + 2s = 2 -(3r + s = 7) 3r + 2s = 2-3r s = -7 0 + s = -5 s = -5
Example 2 continued Solve by the addition-or-subtraction method. 3r + 2s = 2 3r + s = 7 s = -5 3r + s = 7 3r + (-5) = 7 3r 5 = 7 3r = 12 r = 4 The solution is (4, -5).
Example 2 continued Solve by the addition-or-subtraction method. 3r + 2s = 2 3r + s = 7 Check (4, -5) 3r + 2s = 2 3r + s = 7 3(4) + 2(-5) = 2 3(4) + (-5) = 7 12 10 = 2 12 5 = 7 2 =2 7 = 7
Section 9.5 Multiplication with the Addition-or-Subtraction Method
Objective I want to be able to use multiplication with the addition-orsubtraction method to solve systems of linear equations.
Example 1 Solve by the elimination method. 6x + y = 6 3x + 2y = 9 6x + y = 6-2(3x + 2y = 9) 6x + y = 6-6x 4y = -18 0 3y = -12-3y = -12 y = 4
Example 1 continued Solve by the elimination method. 6x + y = 6 3x + 2y = 9 y = 4 6x + y = 6 6x + (4) = 6 6x = 2 x = 2/6 x = 1/3 The solution is (1/3, 4).
Example 1 continued Solve by the elimination method. 6x + y = 6 3x + 2y = 9 Check (1/3, 4) 6x + y = 6 3x + 2y = 9 6(1/3) + 4 = 6 3(1/3) + 2(4) = 9 2 + 4 = 6 1 + 8 = 9 6 = 6 9=9
Example 2 Solve by the elimination method. 2(4s 5t = 3) 4s 5t = 3 3s + 2t = - 15 5(3s + 2t = -15) 8s 10t = 6 15s + 10t = -75 23s + 0 = -69 23s = -69 s = -3
Example 2 continued Solve by the elimination method. 4s 5t = 3 3s + 2t = - 15 s = -3 3s + 2t = -15 3(-3) + 2t = -15-9 + 2t = -15 2t = -6 t = -3 The solution is (-3, -3).
Example 2 continued Solve by the elimination method. 4s 5t = 3 3s + 2t = - 15 Check (-3, -3) 4s 5t = 3 3s + 2t = -15 4(-3) 5(-3) = 3 3(-3) + 2(-3) = -15-12 + 15 = 3-9 6 = -15 3 = 3-15 = - 15
Example 3 Solve by the elimination method. 3x + y = 6 2 x + y = 5 3x + 2y = 12 x + y = 5 3x + 2y = 12-2(x + y = 5) 3x + 2y = 12-2x 2y = -10 x + 0 = 2 x = 2
Example 3 continued Solve by the elimination method. 3x + y = 6 2 x + y = 5 x = 2 x + y = 5 2 + y = 5 y = 3 The solution is (2,3).
Example 3 continued Solve by the elimination method. 3x + y = 6 2 x + y = 5 Check (2, 3) x + y = 5 2 + 3 = 5 5 = 5 3x + y = 6 2 3 2 + 3 = 6 2 6 + 3 = 6 2 3 + 3 = 6 6 = 6
CAUTION! Common Error: Forgetting to Multiply You may forget to multiply both sides of the equal sign by the special number. In order to remember to multiply everything, make sure your work is neat and every step written out.
Try These! Solve, using two equations in two variables. 1. Jeanette has 20 nickels and dimes worth $1.75. How many of each coin does she have? 2. Danny has twice as much money as Connie. Together they have $57. how much money does each have?
Try These! Solve, using two equations in two variables. 1. Jeanette has 20 nickels and dimes worth $1.75. How many of each coin does she have? 5 nickels; 15 dimes 2. Danny has twice as much money as Connie. Together they have $57. how much money does each have? Danny: $38; Connie: $19
And These! Solve, using two equations in two variables. 3. Tickets for the senior play cost $4 for adults and $2 for students. This year there were 600 tickets sold, and the class made $1900. How many of each type of ticket were sold? 4. Kathleen invested $5000, some at 6% and the rest at 5%. Her annual income from the investments is $280. How much is invested at 5%?
And These! Solve, using two equations in two variables. 3. Tickets for the senior play cost $4 for adults and $2 for students. This year there were 600 tickets sold, and the class made $1900. How many of each type of ticket were sold? 350 adult; 250 student 4. Kathleen invested $5000, some at 6% and the rest at 5%. Her annual income from the investments is $280. How much is invested at 5%? $2,000
What about These? ORAL EXERCISES pg. 423 # 3 9 odd
Try These! Solve by the addition-or-subtraction method. 1. x + y = 17 2. a 3b = -1 x y = 1 2a + 3b = 16 3. -3n + 9m = 6 4. 8q + 12r = 20 3n + 4m = 7 5q + 12r = -1 5. 9x 10y = 2 6. a 2b = 0 9x + 2y = -22 a + 2b = 12
Try These! Solve by the addition-or-subtraction method. 1. x + y = 17 2. a 3b = -1 x y = 1 2a + 3b = 16 (9, 8) (5, 2) 3. -3n + 9m = 6 4. 8q + 12r = 20 3n + 4m = 7 5q + 12r = -1 (1, 1) (7, -3) 5. 9x 10y = 2 6. a 2b = 0 9x + 2y = -22 a + 2b = 12 (-2, -2) (6, 3)
And These! Solve each system by using mult. with the addition-or-subtraction method. 1. 4x 2y = 10 2. 5a + 3b = 43 3x y = 12 - a + 7b = -1 3. m + 2n = 6 4. 2s + 3t = 35-2m + 4n = 28 5s 4t = 7 5. 4w 4z = -8-3w + 5z = 0
And These! Solve each system by using mult. with the addition-or-subtraction method. 1. 4x 2y = 10 2. 5a + 3b = 43 3x y = 12 - a + 7b = -1 (7, 9) (8, 1) 3. m + 2n = 6 4. 2s + 3t = 35-2m + 4n = 28 5s 4t = 7 (-4, 5) (7, 7) 5. 4w 4z = -8-3w + 5z = 0 (-5, -3)
TAG! ORAL EXERCISE pg. 427 # 1-9 odd ORAL EXERCISE pg. 431 # 1 9 odd
Clear your calculators! 2 nd + 7 1 2 CLEAR ENTER Take out your agendas! Copy down DUE DATES! Homework: Sections 9.3, 9.4 & 9.5
Journal Entry TOPIC: Elimination Method Answer the following question: Explain the elimination method in your own words.