Final answer:
To solve the system of three linear equations, use the method of elimination. Multiply the equations to eliminate variables and simplify. Solve for the remaining variable, and substitute back to find the values of the other variables.
Step-by-step explanation:
To solve the system of three linear equations, we can use the method of substitution or elimination. Let's use elimination in this case:
First, let's eliminate the variable y by multiplying the first equation by 3 and adding it to the second equation.
3(x-y) + 3x-4z = 3(1) + 12
3x - 3y + 3x - 4z = 3 + 12
6x - 4z = 15
Next, let's eliminate the variable y again by multiplying the first equation by 3 and adding it to the third equation.
3(x-y) + 3y+z = 3(1) + 9
3x - 3y + 3y + z = 3 + 9
3x + z = 12
Now we have a system of two linear equations:
6x - 4z = 15
3x + z = 12
We can solve this system by either substitution or elimination. Let's use elimination again:
Multiply the second equation by 2 to make the coefficients of z the same:
6x - 4z = 15
6x + 2z = 24
Now subtract the new second equation from the first equation:
(6x - 4z) - (6x + 2z) = 15 - 24
-6z = -9
Divide both sides of the equation by -6 to solve for z:
z = 9/6 = 3/2
Substitute the value of z back into one of the original equations to solve for x and y.