# SOLUTION: Solve the following: 3x-5y+2z=19 5x+2y-3z=-8 -2x+3y+5z=7

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 Click here to see ALL problems on Linear-systems Question 72918: Solve the following: 3x-5y+2z=19 5x+2y-3z=-8 -2x+3y+5z=7Answer by rmromero(383)   (Show Source): You can put this solution on YOUR website!``` Solve the following: 3x-5y+2z=19 5x+2y-3z=-8 -2x+3y+5z=7 To solve for systems of 3 linear equation in 3 variable, we use Substitution Method, Elimination Method, Gaussian Reduction method, Cramer's rules. I choose Elimination Method equation 1 : 3x-5y+2z=19 equation 2 : 5x+2y-3z=-8 equation 3 : -2x+3y+5z=7 Let us Eliminate x in equation 1 and equation 2. 3x-5y+2z=19 5x+2y-3z=-8 _____________ Since we cannot eliminate 3x and 5x by adding, we will find a certain number to that will be multiplied to the two equations. Multiply 5 to 3x-5y+2z=19 and -3 to 5x+2y-3z=-8 5(3x-5y+2z=19) -3(5x+2y-3z=-8) _________________ 15x - 25y +10z = 95 -15x - 6y + 9z = 24 Add ____________________ - 31y + 19z = 119 Equation 4 Now eliminate x in equation 1 and equation 3. Multiply 2 to 3x-5y+2z=19 and 3 to -2x+3y+5z=7 2(3x-5y+2z=19) 3(-2x+3y+5z=7) ______________ 6x - 10y + 4z = 38 -6x + 9y + 15z = 21 ____________________ -y + 19z = 59 Equation 5 Then using equation 4 and equation 5 eliminate another variable. I choose z then solve for y. -31y + 19z = 119 -1( - y + 19z = 59 ) ______________________ -31y + 19z = 119 y - 19z = -59 ________________ -30y = 60 y = -2 The value of y will be substituted to either equation 5 or 4 I choose equation 5 -y + 19z = 59, y = -2 -(-2) + 19z = 59 19z = 59 - 2 19z = 57 Divide 19 both sides z = 3 The Substitute y = -2, z = 3 to equation 1 or 2 or 3. I choose equation 3. -2x + 3y + 5z = 7, y = -2, z = 3 -2x + 3(-2) + 5(3) = 7 -2x - 6 + 15 = 7 -2x = -2 x = 1 Checking: equation 1 : 3x-5y+2z=19, x = 1, y = -2, z = 3 3(1)-5(-2)+2(3)=19 3 + 10 + 6 = 19 19 = 19 ---------->True equation 2 : 5x+2y-3z=-8, x = 1, y = -2, z = 3 5(1)+2(-2)-3(3)=-8 5 - 4 - 9 = -8 -8 = -8 ------------>True equation 3 : -2x+3y+5z=7, x = 1, y = -2, z = 3 -2(1)+3(-2)+5(3)= 7 -2 -6 + 15 = 7 7 = 7 ----------> True Therefore, the solution is x = 1, y = -2 and z = 3 ```