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Question 389001: Please help me solve this problem.
Find the solutions for x, y, and z:
x - y - 2z = -1
x + 2y + z = 5
5x +4y - z = 13
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! Any underscore symbol between equations is just spacing between them.
Please email me with any questions at Jennifer_Sadler@stu.southuniversity.edu
x-y-2z=-1,x+2y+z=5,5x+4y-z=13
Move all terms not containing x to the right-hand side of the equation.
x=y+2z-1_x+2y+z=5_5x+4y-z=13
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is y+2z-1.
x=y+2z-1_(y+2z-1)+2y+z=5_5x+4y-z=13
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is y+2z-1.
x=y+2z-1_(y+2z-1)+2y+z=5_5(y+2z-1)+4y-z=13
Remove the parentheses around the expression y+2z-1.
x=y+2z-1_y+2z-1+2y+z=5_5(y+2z-1)+4y-z=13
Since y and 2y are like terms, add 2y to y to get 3y.
x=y+2z-1_3y+2z-1+z=5_5(y+2z-1)+4y-z=13
Since 2z and z are like terms, add z to 2z to get 3z.
x=y+2z-1_3y+3z-1=5_5(y+2z-1)+4y-z=13
Multiply 5 by each term inside the parentheses.
x=y+2z-1_3y+3z-1=5_5y+10z-5+4y-z=13
Since 5y and 4y are like terms, add 4y to 5y to get 9y.
x=y+2z-1_3y+3z-1=5_9y+10z-5-z=13
Since 10z and -z are like terms, add -z to 10z to get 9z.
x=y+2z-1_3y+3z-1=5_9y+9z-5=13
Move all terms not containing y to the right-hand side of the equation.
x=y+2z-1_3y=-3z+1+5_9y+9z-5=13
Add 5 to 1 to get 6.
x=y+2z-1_3y=-3z+6_9y+9z-5=13
Divide each term in the equation by 3.
x=y+2z-1_(3y)/(3)=-(3z)/(3)+(6)/(3)_9y+9z-5=13
Simplify the left-hand side of the equation by canceling the common factors.
x=y+2z-1_y=-(3z)/(3)+(6)/(3)_9y+9z-5=13
Simplify the right-hand side of the equation by simplifying each term.
x=y+2z-1_y=-z+2_9y+9z-5=13
Replace all occurrences of y with the solution found by solving the last equation for y. In this case, the value substituted is -z+2.
x=y+2z-1_y=-z+2_9(-z+2)+9z-5=13
Multiply 9 by each term inside the parentheses.
x=y+2z-1_y=-z+2_-9z+18+9z-5=13
Since -9z and 9z are like terms, subtract 9z from -9z to get 0.
x=y+2z-1_y=-z+2_0+18-5=13
Subtract 5 from 18 to get 13.
x=y+2z-1_y=-z+2_13=13
Since 13=13, the equation will always be true.
x=y+2z-1_y=-z+2_Equations are Always True
On this equation, you could also add all three together, instead of using the method of substitution and follow the instructions below (I already added them together)
7x+5y-2z=17
Move all terms not containing y to the right-hand side of the equation.
5y=-7x+2z+17
Divide each term in the equation by 5.
(5y)/(5)=-(7x)/(5)+(2z)/(5)+(17)/(5)
Simplify the left-hand side of the equation by canceling the common factors.
y=-(7x)/(5)+(2z)/(5)+(17)/(5)
Simplify the right-hand side of the equation by simplifying each term.
y=(-7x+2z+17)/(5)
Now we have solved for y, we can do x.
7x+5y-2z=17
Move all terms not containing x to the right-hand side of the equation.
7x=-5y+2z+17
Divide each term in the equation by 7.
(7x)/(7)=-(5y)/(7)+(2z)/(7)+(17)/(7)
Simplify the left-hand side of the equation by canceling the common factors.
x=-(5y)/(7)+(2z)/(7)+(17)/(7)
Simplify the right-hand side of the equation by simplifying each term.
x=(-5y+2z+17)/(7)
and now z.....
7x+5y-2z=17
Move all terms not containing z to the right-hand side of the equation.
-2z=-7x-5y+17
Divide each term in the equation by -2.
-(2z)/(-2)=-(7x)/(-2)-(5y)/(-2)+(17)/(-2)
Simplify the left-hand side of the equation by canceling the common factors.
z=-(7x)/(-2)-(5y)/(-2)+(17)/(-2)
Simplify the right-hand side of the equation by simplifying each term.
z= (7x-17)/(2)+(5y)/(2)
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