SOLUTION: Find the values of x,y and z which make all these equations true: Y+z=78 , z+x= 70 and x+y=54

Question 891960: Find the values of x,y and z which make all these equations true:
Y+z=78 , z+x= 70 and x+y=54
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you have:
y + z = 78 (equation 1)
x + z = 70 (equation 2)
x + y = 54 (equation 3)

start with equation 1.
solve for y to get y = 78 - z

start with equation 2.
solve for x to get x = 70 - z

start with equation 3.
replace x with 70 - z
replace y with 78 - z
equation becomes 70 - z + 78 - z = 54
combine like terms to get 148 - 2z = 54
add 2z to both sides of the equation and subtract 54 from both sides of the equation to get 94 = 2z
solve for z to get z = 47

start with equation 2 to get:

x + z = 70
replace z with 47
equation becomes x + 47 = 70
subtract 47 from both sides of the equation to get x = 23

so far you have x = 23 and z = 47

start with equation 1 to get:

y + z = 78
replace z with 47 to get y + 47 = 78
subtract 47 from both sides of the equation to get y = 31

you now have x = 23, y = 31, z = 47
those are your solutions.
replace x,y,z in the original equations and they should hold true.

y + z becomes 31 + 47 = 78 which is true.
x + z becomes 23 + 47 = 70 which is true.
x + y becomes 23 + 31 = 54 which is true.

you did good.

another way to solve it is to treat it like 3 sets of equations that need to be solved simultaneously.

the 3 equations are:

y + z = 78 (equation 1)
x + z = 70 (equation 2)
x + y = 54 (equation 3)

take the first 2 equations and solve them simultaneously to remove z.

start with:

y + z = 78
x + z = 70

subtract second equation from first equation to get y - x = 8 (equation 4)

take that and solve it simultaneously with equation 3 to get:

x + y = 54 (equation 3)
-x + y = 8 (equation 4)

add the 2 equations together to get 2y = 62
solve for y to get y = 31

so far you have y = 31

go back to equation 1 to get:

y + z = 78 (equation 1)
replace y with 31 to get:|
31 + z = 78
solve for z to get z = 47

so far you have y = 31 and z = 47

go back to equation 2 to get:

x + z = 70 (equation 2)
replace z with 47 to get:
x + 47 = 70
solve for x to get x = 23

now you have x = 23, y = 31, z = 47

those are your solutions as before.

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