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Systems of 3-Variable Equations

Systems of 3-Variable Equations. Mei Huang, Nahom Ghile, Jonathan Ye. A system of 2 variable equations is the intersection of two lines . A system of 3 variable equations is the intersection of three planes.

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Systems of 3-Variable Equations

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  1. Systems of 3-Variable Equations Mei Huang, Nahom Ghile, Jonathan Ye

  2. A system of 2 variable equationsis theintersection oftwo lines. A system of 3 variable equations is theintersection of three planes.

  3. An example of a three variable equation would be x - y + z = 2. The equation has three variables x, y, and z. A three variable system would then be x - y + z = 2 2x + y + 4z = 4 -x + 3y - z = 6 There are three equations to the system and each equation has three variables. These systems would have to be solved differently from regular two variable systems

  4. Solving With Cramer's Rule x - y + z = 2 2x + y + 4z = 4 -x + 3y - z = 6 1. find determinant of coefficient matrix. 1 -1 1 1 -1 -1+4+6 = 9 D 2 1 4 2 1 -1+12+2 = 13 -1 3 -1 -1 3 9 - 13 = D=-4 2.find the determinant of x, y and, z variable by replacing each of the x, y, or z coefficients with the constants. 2 -1 1 2 -1 Dx 4 1 4 4 1 Dx=-48 6 3 -1 6 3 1 2 1 1 2 Dy 2 4 4 2 4 Dy=-16 -1 6 -1 -1 6 1 -1 2 1 -1 Dz 2 1 4 2 1 Dz=24 -1 3 6 -1 3 3.now, to find x, y, and z, divide Dx, Dy, and Dz each by D. Dx = -48 = D -4 Dy = -16 = D -4 Dz = 24 = D -4 x=12 y=4 z=-6

  5. Solving With Inverse Matrices on Calculator x-y+z = 2 2x+y+4z = 4 -x+3y - z = 6 1.First go to2nd x ^-1. 2. Scroll to edit and scroll down to A or the first open Matrix and hit enter. 3.Change the dimensions to 3x3. Then enter the coefficients in corresponding order. 1 -1 1 2 1 4 -1 3 -1 6.Go to 2nd x^-1 scroll down and hit A. Then hit x^-1. Then repeat the first sentence but hit B. Then hit enter. Answers will be matrix from first variable down. 4. Repeat step 2 except this time going to B. 5. Change the dimensions to 3x1 and enter the answers in corresponding order. 2 4 6 -1 x = 12 y = 4 z = -6 A B 12 4 -6

  6. Word Problem Example Ex: There are 3 types of canned vegetables: tomatoes,corn, and peas. James wanted to how much each can weighed. He finds that the total weight of all three cans is 6 oz. Two cans of tomatoes plus on can of corn and one can of peas equal to 8 oz. And one can of tomatoes plus two cans of corn plus one can of peas equals 7 oz. Find the weight of each canned vegetable. 1. Label each variable: 2. write information into 3 x=weight of cans of tomatoes equations: y=weight of cans of corn x+y+z = 6 z=weight of cans of peas 2x+y+z = 8 x+2y+z = 7 3. put in matrices 1 1 1 2 1 1 1 2 1 A B 4. Solve (A^-1)x(B)= 2 =x 1 =y 3 =z 6 8 7 The can of tomatoes weigh 2 ounces, the can of corn weigh 1 ounce and the can of peas weigh 3 ounces.

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