To solve the system of equations, let's start with the equations you provided:
- ( x - y = 3y )
- ( z - 2y = y )
- ( z = 5 )
First, we'll simplify the equations:
From the first equation, rearranging gives:
[
x - y = 3y \implies x = 4y
]
From the second equation, rearranging gives:
[
z - 2y = y \implies z = 3y
]
Now we can substitute the value of ( z ) from equation 3 into the rearranged second equation:
[
3y = 5
]
Solving for ( y ):
[
y = \frac{5}{3}
]
Now that we have ( y ), we can substitute it into the equation for ( z ):
[
z = 3y = 3 \cdot \frac{5}{3} = 5
]
This is consistent with the given ( z = 5 ).
Next, we substitute ( y ) into the equation for ( x ):
[
x = 4y = 4 \cdot \frac{5}{3} = \frac{20}{3}
]
So, we have determined the values of the variables:
- ( x = \frac{20}{3} )
- ( y = \frac{5}{3} )
- ( z = 5 )
The solution to the system of equations is:
[
(x, y, z) = \left(\frac{20}{3}, \frac{5}{3}, 5\right)
]
