A2H Assignment #8 Cramer s Rule Unit 2: Matrices and Systems Name: DUE Date: Friday 12/2 as a Packet What is the Cramer s Rule used for? à Another method to solve systems that uses matrices and determinants. When can Cramer s Rule be used? à When there are two unknown variables and two equations or when there are three unknown variables and three equations. Therefore, whenever you can build your system into a square 2x2 or square 3x3 coefficient matrix, Cramer s rule can be used. What is a coefficient matrix? à For example LINEAR SYSTEM COEFFICIENT MATRIX ax + by e a b cx + dy f c d 3x 2y 10 5x + 3y 4 3 2 5 3 If I have to use determinates, what is the formula again? à DETERMINANT OF A 2 ª 2 MATRIX a c b d a b c d det ad º cb The determinant of a 2 ª 2 matrix is the difference of the products of the entries on the diagonals. DETERMINANT OF A 3 ª 3 MATRIX 1 Repeat the first two columns to the right of the determinant. 2 Subtract the sum of the products in red from the sum of the products in blue. a d g b e h c f i det a b c a b d e (aei + bfg + cdh) º (gec + hfa + idb) d e f g h i g h GOAL 2 USING CRAMER S RULE You can use determinants to solve a system of linear equations. The method, and named after the Swiss mathematician Gabriel Cramer (1704º1752), uses the coefficient matrix of the linear system. The Cramer s Rule called for Cramer s 2x2 rule Systemsà LINEAR SYSTEM ax + by e cx + dy f COEFFICIENT MATRIX a c b d CRAMER S RULE FOR A 2 X 2 SYSTEM Let A be the coefficient matrix of this linear system: ax + by e cx + dy f If 0, then the system has exactly one solution. The solution is: f e b a e d c f x and y In Cramer s rule, notice that the denominator for x and y is the determinant of the coefficient matrix of the system. The numerators for x and y are the determinants of the matrices formed by using the column of constants as replacements for the coefficients of x and y, respectively. EXAMPLE 4 Using Cramer s Rule for a 2 2 System STUDENT HELP Use Cramer s rule to solve this system: 8x + 5y 2 2x º 4y º10
Example (step-by-step): Solve the system below by Cramer s rule. 2x + 5y 11 3x 4y 28 Step 1: Write the coefficient matrix and find its determinant Coefficient Matrix: A 8 15 Step 2: To set up the formula for x, swap out the x values (first column) and replace them with the solution values (11 and 28). Take the determinant of this divided by the. x Step 3: Calculate the needed values for x. x 44 140 184 8 Step 4: To set up the formula for y, swap out the y values (second column) and replace them with the solution values (11 and 28). Take the determinant of this divided by the. y Step 5: Calculate the needed values for y. y Your answer is : (8, -1) 56 33 23 1 Problem 1 (step-by-step): Solve the system below by Cramer s rule. 8x + 5y 2 2x 4y 10
Step 1: Write the coefficient matrix and find its determinant Coefficient Matrix: A Step 2: To set up the formula for x, swap out the x values (first column) and replace them with the solution values. Take the determinant of this divided by the. x Step 3: Calculate the needed values for x. Step 4: To set up the formula for y, swap out the y values (second column) and replace them with the solution values. Take the determinant of this divided by the. y Step 5: Calculate the needed values for y. Your answer is :
Problem 2: Solve the system below by Cramer s rule. 3x 5y 26 5x + y 6 The Cramer s Rule for 3x3 Systemsà Let A be the coefficient matrix of this linear system: ax + by + cz j dx + ey + fz k gx + hy + iz l If 0, then the system has exactly one solution. The solution is: j b c a j c a b j k e f d k f d e k l h i g l i g h l x, y, and z Example (step-by-step): Solve the system below by Cramer s rule. x 5y + z 17 5x 5y + 5z 5 2x + 5y 3z 10 Step 1: Write the coefficient matrix and find its determinant Coefficient Matrix: A 1 5 5 5 ( 15 + 50 + 25) ( 10 + 25 + 75) 90 ( 110) 20
Step 2: To set up the formula for x, swap out the x values (first column) and replace them with the solution values (17, 5, -10). Take the determinant of this divided by the. x 17 5 1 5 5 5 10 5 3 17 5 1 5 5 5 10 5 3 17 5 1 5 5 5 10 5 3 Step 3: Calculate the needed values for x. 17 5 1 5 5 5 10 5 3 17 5 5 5 10 5 20 (255 + 250 + 25) (50 + 425 + 75) 530 550 20 x 20 20 20 1 Step 4: To set up the formula for y, swap out the y values (second column) and replace them with the solution values (17, 5, -10). Take the determinant of this divided by the. y 1 17 1 5 5 5 2 10 3 1 17 1 5 5 5 2 10 3 1 17 1 5 5 5 2 10 3 Step 5: Calculate the needed values for y. 1 17 1 5 5 5 2 10 3 1 17 5 5 2 10 20 (15 +170 + 50) (10 + 50 + 225) 235 285 60 y 60 60 20 3 Step 6: To set up the formula for z, swap out the z values (third column) and replace them with the solution values (17, 5, -10). Take the determinant of this divided by the. z 7 10 7 10 7 10 Step 7: Calculate the needed values for z. 7 10 1 5 5 5 z 80 80 20 4 Your answer is : (-1, -3, -4) 20 ( 50 + 50 + 425) ( 170 + 25 + 250) 525 ( 445) 80
Problem 3 (step-by-step): Given the linear equations below, solve for the values of x, y, and z by Cramer s Rule. x y 2z 6 3x + 2y 25 4x + y z 12 Step 1: Write the coefficient matrix and find its determinant Coefficient Matrix: A ( + + ) ( + + ) Step 2: To set up the formula for x, swap out the x values (first column) and replace them with the solution values. Take the determinant of this divided by the. x Step 3: Calculate the needed values for x. ( + + ) ( + + ) x
Step 4: To set up the formula for y, swap out the y values (second column) and replace them with the solution values. Take the determinant of this divided by the. y Step 5: Calculate the needed values for y. ( + + ) ( + + ) y Step 6: To set up the formula for z, swap out the z values (third column) and replace them with the solution values. Take the determinant of this divided by the. z Step 7: Calculate the needed values for z. ( + + ) ( + + ) z Your answer is
Problem 4: Given the linear equations below, solve for the values of x, y, and z by Cramer s Rule. x + y z 4 3x + 2y + 4z 17 x + 5y + z 8