x+y+z=30
x-y-z=8
x-y+2z=17
x is the same in all equations
y is the same in all equations
z is the same in all equations
Add the first and the second ---> 2x = 39
x = 19
add the first and the third ----> 2x + 3z = 47
38 + 3z = 47
z = 3
now put those values back into the second
19 - y - 3 = 8
y = 8
i am 50 years old, i am sorry but i need this spelled out for me if possible
Well Steve, I am 68 years old, lol
I assume you are solving these equations?
x + y + z = 30 ---- first equation
x - y - z = 8 -----second equation
2x + 0 + 0 = 38 ---- I added them
x = 19 divided by 2, we now know x
x + y + z = 30 --- first equation again
x - y + 2z = 17 --third equation
2x + 0 + 3z = 47 ---- added them
2x + 3z = 47 , but we know x = 19 so..
38 + 3z = 47
3z = 47 - 38
3z = 9
z = 3 , so now we have x=19 and z=3
let's put this back into the first, or any one really
x + y + z = 30
19 + y + 3 = 30
y + 22 = 30
y = 30-22
y = 8
so x=19, y=8, and z=3
notice those values work in all 3 equations.
Reiny,thanks so much for your help on this!!!!!!!!!!!
To find the values of x, y, and z, we can use the method of solving a system of linear equations.
First, let's rearrange the equations in a form that we can use to eliminate variables.
1) x + y + z = 30
2) x - y - z = 8
3) x - y + 2z = 17
To eliminate the variables y and z, we can add equations 2 and 3 together.
(x - y - z) + (x - y + 2z) = 8 + 17
2x - 2y + z = 25 (Equation 4)
Now, we can eliminate the variable y by subtracting equation 4 from equation 1.
(x + y + z) - (2x - 2y + z) = 30 - 25
x + 3y = 5 (Equation 5)
We now have two equations with two variables remaining: equation 5 and equation 4.
Let's solve these equations simultaneously.
We'll multiply equation 5 by 2 and equation 4 by 3 to eliminate the variable y.
2(x + 3y) = 2(5)
3(2x - 2y + z) = 3(25)
2x + 6y = 10 (Equation 6)
6x - 6y + 3z = 75 (Equation 7)
Adding equation 6 and equation 7 together allows us to eliminate the variable y.
(2x + 6y) + (6x - 6y + 3z) = 10 + 75
8x + 3z = 85 (Equation 8)
Now we have two equations with two variables remaining: equation 8 and equation 4.
To eliminate the variable z, we'll multiply equation 8 by 2 and subtract equation 4 from it.
2(8x + 3z) - (2x - 2y + z) = 2(85) - 25
16x + 6z - 2x + 2y - z = 170 - 25
14x + 5z + 2y = 145 (Equation 9)
We only have one variable left, which is x.
Finally, we can isolate x in equation 9 by subtracting equation 5 from it.
(14x + 5z + 2y) - (x + 3y) = 145 - 5
14x + 5z + 2y - x - 3y = 140
13x + 5z - y = 140 (Equation 10)
Since the value of x is the same in all equations, we can solve this simplified equation to find the value of x.
From equation 10: 13x + 5z - y = 140
Since the values of y and z are the same in all equations, we can substitute them with y and z values from any equation. Let's use equation 1.
From equation 1: x + y + z = 30
Rearranging this equation: x = 30 - (y + z)
Now substitute x and rearrange the equation from equation 10:
13(30 - (y + z)) + 5z - y = 140
390 - 13y - 13z + 5z - y = 140
-14y - 8z = -250
From this equation, we can see that the values of y and z are dependent on each other. There isn't a unique solution for y and z.
However, once you have the values of y and z, you can substitute them back into any of the original equations to solve for x.