Question 391152: 2x+y-z=7
2x-y+3z=13
3x+2y-z=17
Found 2 solutions by haileytucki, richard1234:
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! Please state if these are SEPERATE because they can be solved by the substitution method:
2x+y-z=7_2x-y+3z=13_3x+2y-z=17
Move all terms not containing y to the right-hand side of the equation.
y=-2x+z+7_2x-y+3z=13_3x+2y-z=17
Replace all occurrences of y with the solution found by solving the last equation for y. In this case, the value substituted is -2x+z+7.
y=-2x+z+7_2x-(-2x+z+7)+3z=13_3x+2y-z=17
Replace all occurrences of y with the solution found by solving the last equation for y. In this case, the value substituted is -2x+z+7.
y=-2x+z+7_2x-(-2x+z+7)+3z=13_3x+2(-2x+z+7)-z=17
Multiply -1 by each term inside the parentheses.
y=-2x+z+7_2x+2x-z-7+3z=13_3x+2(-2x+z+7)-z=17
Since 2x and 2x are like terms, add 2x to 2x to get 4x.
y=-2x+z+7_4x-z-7+3z=13_3x+2(-2x+z+7)-z=17
Since -z and 3z are like terms, subtract 3z from -z to get 2z.
y=-2x+z+7_4x+2z-7=13_3x+2(-2x+z+7)-z=17
Multiply 2 by each term inside the parentheses.
y=-2x+z+7_4x+2z-7=13_3x-4x+2z+14-z=17
Since 3x and -4x are like terms, add -4x to 3x to get -x.
y=-2x+z+7_4x+2z-7=13_-x+2z+14-z=17
Since 2z and -z are like terms, add -z to 2z to get z.
y=-2x+z+7_4x+2z-7=13_-x+z+14=17
Move all terms not containing x to the right-hand side of the equation.
y=-2x+z+7_4x+2z-7=13_-x=-z-14+17
Add 17 to -14 to get 3.
y=-2x+z+7_4x+2z-7=13_-x=-z+3
Multiply each term in the equation by -1.
y=-2x+z+7_4x+2z-7=13_-x*-1=-z*-1+3*-1
Multiply -x by -1 to get x.
y=-2x+z+7_4x+2z-7=13_x=-z*-1+3*-1
Simplify the right-hand side of the equation by multiplying out all the terms.
y=-2x+z+7_4x+2z-7=13_x=z-3
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is z-3.
y=-2x+z+7_4(z-3)+2z-7=13_x=z-3
Multiply 4 by each term inside the parentheses.
y=-2x+z+7_4z-12+2z-7=13_x=z-3
Since 4z and 2z are like terms, add 2z to 4z to get 6z.
y=-2x+z+7_6z-12-7=13_x=z-3
Subtract 7 from -12 to get -19.
y=-2x+z+7_6z-19=13_x=z-3
Since -19 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 19 to both sides.
y=-2x+z+7_6z=19+13_x=z-3
Add 13 to 19 to get 32.
y=-2x+z+7_6z=32_x=z-3
Divide each term in the equation by 6.
y=-2x+z+7_(6z)/(6)=(32)/(6)_x=z-3
Simplify the left-hand side of the equation by canceling the common factors.
y=-2x+z+7_z=(32)/(6)_x=z-3
Simplify the right-hand side of the equation by simplifying each term.
y=-2x+z+7_z=(16)/(3)_x=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 (16)/(3).
y=-2x+((16)/(3))+7_z=(16)/(3)_x=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 (16)/(3).
y=-2x+((16)/(3))+7_z=(16)/(3)_x=((16)/(3))-3
Remove the parentheses around the expression (16)/(3).
y=-2x+(16)/(3)+7_z=(16)/(3)_x=((16)/(3))-3
To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 3. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
y=-2x+7*(3)/(3)+(16)/(3)_z=(16)/(3)_x=((16)/(3))-3
Complete the multiplication to produce a denominator of 3 in each expression.
y=-2x+(21)/(3)+(16)/(3)_z=(16)/(3)_x=((16)/(3))-3
Combine the numerators of all fractions that have common denominators.
y=-2x+(21+16)/(3)_z=(16)/(3)_x=((16)/(3))-3
Add 16 to 21 to get 37.
y=-2x+(37)/(3)_z=(16)/(3)_x=((16)/(3))-3
Remove the parentheses around the expression (16)/(3).
y=-2x+(37)/(3)_z=(16)/(3)_x=(16)/(3)-3
To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 3. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
y=-2x+(37)/(3)_z=(16)/(3)_x=-3*(3)/(3)+(16)/(3)
Complete the multiplication to produce a denominator of 3 in each expression.
y=-2x+(37)/(3)_z=(16)/(3)_x=-(9)/(3)+(16)/(3)
Combine the numerators of all fractions that have common denominators.
y=-2x+(37)/(3)_z=(16)/(3)_x=(-9+16)/(3)
Add 16 to -9 to get 7.
y=-2x+(37)/(3)_z=(16)/(3)_x=(7)/(3)
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is (7)/(3).
y=-2((7)/(3))+(37)/(3)_z=(16)/(3)_x=(7)/(3)
Multiply -2 by each term inside the parentheses.
y=-(14)/(3)+(37)/(3)_z=(16)/(3)_x=(7)/(3)
Complete the multiplication to produce a denominator of 3 in each expression.
y=(37)/(3)-(14)/(3)_z=(16)/(3)_x=(7)/(3)
Combine the numerators of all fractions that have common denominators.
y=(37-14)/(3)_z=(16)/(3)_x=(7)/(3)
Subtract 14 from 37 to get 23.
y=(23)/(3)_z=(16)/(3)_x=(7)/(3)
This is the solution to the system of equations.
y=(23)/(3)_z=(16)/(3)_x=(7)/(3)
Answer by richard1234(7193)  (Show Source):
You can put this solution on YOUR website! Add the first two equations (this clears out the y-term) to get
4x + 2z = 20 --> 2x + z = 10
Multiply the second equation by 2, so we can add the second and third equations and also clear the y-terms. This results in another equation in terms of x and z that is not equivalent to the one we already obtained.
4x - 2y + 6z = 26
3x + 2y - z = 17
--------------------
7x + 5z = 43
We can rewrite this as 5(2x + z) - 3x = 43 --> 5(10) - 3x = 43 --> x = 7/3, z = 16/3. Substituting x and z into any of the equations, we get y = 23/3. Therefore the solution (x,y,z) is (7/3, 23/3, 16/3).
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