Determinants WKST 9102021 7 28 PM 3 7

Determinants WKST 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 1

Cramer’s Rule section 4. 4 dangjim@gmail. com Revised © 2011 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 2

For Your Information • Named by Gabriel Cramer in 1700’s • Uses determinant to find solution of the system – Revisit the formula of the determinant for 2 x 2 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 3

Recall Equation of a determinant (NOT THE INVERSE): After Death – Before Christ 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 4

Steps A. Find the determinant B. To get x: 1. Write in determinant form (without the equal sign) 2. REPLACE the x-coordinates with the numbers associated after the equal sign 3. Divide it by the determinant C. To get y: 1. Same process as x but replace it with the y-coordinate D. Write the answer as an ordered pair 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 5

Equations To get x: D = Determinant Equation: To get y: . 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 6

Example 1 Use Cramer’s Rule to determine this system: What is the determinant? 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 7

Example 1 Use Cramer’s Rule to determine this system: First, setup the X solution 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 8

Example 1 Use Cramer’s Rule to determine this system: First, setup the X solution To get X, get rid of the coefficients in front of X and replace them with the answers 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 9

Example 1 Use Cramer’s Rule to determine this system: Solve for X. 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 10

Example 1 Use Cramer’s Rule to determine this system: Solve for X. 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 11

Example 1 Use Cramer’s Rule to determine this system: Finally, put D in and simplify 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 12

Example 1 Use Cramer’s Rule to determine this system: First, setup the Y solution 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 13

Example 1 Use Cramer’s Rule to determine this system: First, setup the Y solution To get Y, get rid of the coefficients in front of Y and replace them with the answers 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 14

Example 1 Use Cramer’s Rule to determine this system: Solve for Y. 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 15

Example 1 Use Cramer’s Rule to determine this system: Solve for Y. 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 16

Example 1 Use Cramer’s Rule to determine this system: Put D in and simplify 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 17

Example 1 Use Cramer’s Rule to determine this system: 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 18

Example 2 Use Cramer’s Rule to determine this system: 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 19

Your Turn Use Cramer’s Rule to determine this system: 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 20

Special Occurrences No Solution: Infinite Number of Solutions: 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 21

Example 3 Use Cramer’s Rule to determine this system: 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 22

Your Turn Use Cramer’s Rule to determine this system: 9/10/2021 7: 28 PM 3. 7 a - Cramer's Rule 23

Example 4 On Monday, Angela babysat for 4 hours, did yard work for 2 hours, and earned a total of $41. On Friday, she babysat for 5 hours, did yard work for 3 hours, and earned a total of $65. Use Cramer’s Rule to determine the hourly rate for each activity. x = babysitting, y = yard work 9/10/2021 7: 29 PM 3. 7 a - Cramer's Rule 24

Example 4 On Monday, Angela babysat for 4 hours, did yard work for 2 hours, and earned a total of $41. On Friday, she babysat for 5 hours, did yard work for 3 hours, and earned a total of $65. Use Cramer’s Rule to determine the hourly rate for each activity. 9/10/2021 7: 29 PM 3. 7 a - Cramer's Rule 25

Example 5 A nutritionist creates a diet for a long-distance runner that includes 3400 Calories from 680 grams of food, with half the Calories coming from carbohydrates. How many grams of protein, carbohydrates, and fat will this diet include? 4 p + 4 c + 9 f = 3400 p + c + f = 680 4 c = 1700 9/10/2021 7: 29 PM Equation for total Calories Total grams of food Calories from carbohydrates, 3. 7 a - Cramer's Rule 26

Example 5 A nutritionist creates a diet for a long-distance runner that includes 3400 Calories from 680 grams of food, with half the Calories coming from carbohydrates. How many grams of protein, carbohydrates, and fat will this diet include? p = 119 c = 725 f = 136 The diet includes 119 grams of protein, 425 grams of carbohydrates, and 136 grams of fat. 9/10/2021 7: 29 PM 3. 7 a - Cramer's Rule 27

Assignment Cramer’s Rule WKST 9/10/2021 7: 29 PM 3. 7 a - Cramer's Rule 28