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Question 389000: Please help me solve this problem.
Find the solutions for x, y, and z:
x + 2y - z = 5
2x + y + z = 1
x - y + z = -1
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! x+2y-z=5,2x+y+z=1,x-y+z=-1
Move all terms not containing x to the right-hand side of the equation.
x=-2y+z+5_2x+y+z=1_x-y+z=-1
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is -2y+z+5.
x=-2y+z+5_2(-2y+z+5)+y+z=1_x-y+z=-1
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is -2y+z+5.
x=-2y+z+5_2(-2y+z+5)+y+z=1_(-2y+z+5)-y+z=-1
Multiply 2 by each term inside the parentheses.
x=-2y+z+5_-4y+2z+10+y+z=1_(-2y+z+5)-y+z=-1
Since -4y and y are like terms, subtract y from -4y to get -3y.
x=-2y+z+5_-3y+2z+10+z=1_(-2y+z+5)-y+z=-1
Since 2z and z are like terms, add z to 2z to get 3z.
x=-2y+z+5_-3y+3z+10=1_(-2y+z+5)-y+z=-1
Remove the parentheses around the expression -2y+z+5.
x=-2y+z+5_-3y+3z+10=1_-2y+z+5-y+z=-1
Since -2y and -y are like terms, subtract y from -2y to get -3y.
x=-2y+z+5_-3y+3z+10=1_-3y+z+5+z=-1
Since z and z are like terms, add z to z to get 2z.
x=-2y+z+5_-3y+3z+10=1_-3y+2z+5=-1
Move all terms not containing y to the right-hand side of the equation.
x=-2y+z+5_-3y=-3z-10+1_-3y+2z+5=-1
Add 1 to -10 to get -9.
x=-2y+z+5_-3y=-3z-9_-3y+2z+5=-1
Divide each term in the equation by -3.
x=-2y+z+5_-(3y)/(-3)=-(3z)/(-3)-(9)/(-3)_-3y+2z+5=-1
Simplify the left-hand side of the equation by canceling the common factors.
x=-2y+z+5_y=-(3z)/(-3)-(9)/(-3)_-3y+2z+5=-1
Simplify the right-hand side of the equation by simplifying each term.
x=-2y+z+5_y=z+3_-3y+2z+5=-1
Replace all occurrences of y with the solution found by solving the last equation for y. In this case, the value substituted is z+3.
x=-2y+z+5_y=z+3_-3(z+3)+2z+5=-1
Multiply -3 by each term inside the parentheses.
x=-2y+z+5_y=z+3_-3z-9+2z+5=-1
Since -3z and 2z are like terms, subtract 2z from -3z to get -z.
x=-2y+z+5_y=z+3_-z-9+5=-1
Add 5 to -9 to get -4.
x=-2y+z+5_y=z+3_-z-4=-1
Since -4 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 4 to both sides.
x=-2y+z+5_y=z+3_-z=4-1
Subtract 1 from 4 to get 3.
x=-2y+z+5_y=z+3_-z=3
Multiply each term in the equation by -1.
x=-2y+z+5_y=z+3_-z*-1=3*-1
Multiply -z by -1 to get z.
x=-2y+z+5_y=z+3_z=3*-1
Multiply 3 by -1 to get -3.
x=-2y+z+5_y=z+3_z=-3
Replace all occurrences of z with the solution found by solving the last equation for z. In this case, the value substituted is -3.
x=-2y+(-3)+5_y=z+3_z=-3
Replace all occurrences of z with the solution found by solving the last equation for z. In this case, the value substituted is -3.
x=-2y+(-3)+5_y=(-3)+3_z=-3
Remove the parentheses around the expression -3.
x=-2y-3+5_y=(-3)+3_z=-3
Add 5 to -3 to get 2.
x=-2y+2_y=(-3)+3_z=-3
Remove the parentheses around the expression -3.
x=-2y+2_y=-3+3_z=-3
Add 3 to -3 to get 0.
x=-2y+2_y=0_z=-3
Replace all occurrences of y with the solution found by solving the last equation for y. In this case, the value substituted is 0.
x=-2(0)+2_y=0_z=-3
Multiply -2 by each term inside the parentheses.
x=0+2_y=0_z=-3
Remove the 0 from the polynomial; adding or subtracting 0 does not change the value of the expression.
x=2_y=0_z=-3
This is the solution to the system of equations.
x=2_y=0_z=-3
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