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Question 1151846: 7x − y − z = 34
x − 3y + z = 6
x + 2y − z = 4
This is my first question! Find, x,y,z and make sure that they are all true!
4a − 3b = 1
6a − 8c = 1
2b − 4c = 0
This is my second question! find, x,y,z and make sure that they are all true!
Found 2 solutions by Edwin McCravy, jim_thompson5910:
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website!
There are bunches of different ways to solve systems of equations.
Which method are you studying?
7x − y − z = 34
x − 3y + z = 6
x + 2y − z = 4
Here are 8 different solutions to that system. If you substitute any one of
them, all three equations will be true.
Solution 1: x = 1 y = -8 z = -19
Solution 2: x = 2 y = -6 z = -14
Solution 3: x = 3 y = -4 z = -9
Solution 4: x = 4 y = -2 z = -4
Solution 5: x = 5 y = 0 z = 1
Solution 6: x = 6 y = 2 z = 6
Solution 7: x = 7 y = 4 z = 11
Solution 8: x = 8 y = 6 z = 16
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4a − 3b = 1
6a − 8c = 1
2b − 4c = 0
Here's one solution to that system. Substitute it and you'll get a true
equation:
Edwin
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
I'll do the first problem to get you started
Given system of equations
7x − y − z = 34 ..... equation (1)
x − 3y + z = 6 ..... equation (2)
x + 2y − z = 4 ..... equation (3)
Solve equation (1) for z
7x - y - z = 34
7x - y - z+z = 34+z
7x - y = 34+z
7x - y - 34 = 34+z - 34
7x - y - 34 = z
z = 7x - y - 34 .... call this equation (4)
Plug this into equation (2). Simplify. Then solve for y.
x - 3y + z = 6
x - 3y + 7x - y - 34 = 6
8x - 4y - 34 = 6
8x - 4y = 40
2x - y = 10
y = 2x - 10 ... we'll use this later.
Move onto equation (3) and plug in z = 7x - y - 34. Simplify
x + 2y - z = 4
x + 2y - (7x - y - 34) = 4
x + 2y - 7x + y + 34 = 4
-6x + 3y + 34 = 4
-6x + 3y = -30
-2x + y = -10
Next replace y with 2x-10, which was from y = 2x-10 found earlier
-2x + y = -10
-2x + 2x - 10 = -10
-10 = -10
Both sides are the same value.
We have a consistent and dependent system.
There are infinitely many solutions
Go back to equation (4). Plug in y = 2x-10
z = 7x - y - 34
z = 7x - (2x - 10) - 34
z = 7x - 2x + 10 - 34
z = 5x - 24
We have now expressed z in terms of x
The variable y is in terms of x as well
y = 2x - 10
We can let x be the free variable. It is anything you want while y and z depend on what x is
x = any number
y = 2x-10
z = 5x-24
One way to express all the solutions is to write
(x, y, z) = (x, 2x-10, 5x-24)
This x,y,z triple will allow you to generate any solution you want
For instance, if x = 0, then
(x, y, z) = (x, 2x-10, 5x-24)
(x, y, z) = (0, 2*0-10, 5*0-24)
(x, y, z) = (0, -10, -24)
which is one solution
Or if x = 7, then
(x, y, z) = (x, 2x-10, 5x-24)
(x, y, z) = (7, 2*7-10, 5*7-24)
(x, y, z) = (7, 4, 11)
which is another solution.
Or if x = 5, then
(x, y, z) = (x, 2x-10, 5x-24)
(x, y, z) = (5, 2*5-10, 5*5-24)
(x, y, z) = (5, 0, 1)
which is another solution.
This process goes on forever because x can be any real number (and there are infinitely many of those).
note: x does not have to be the free variable. We could make either y or z have that role. I would ask your teacher which s/he prefers to have as the free variable. Often times, it is sufficient enough to say "infinitely many solutions" and not have to specify the solution form.
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