Answer:
So, the solution to the system of equations is:
x = -16
y = -4 - x = -4 - (-16) = 12
z = 10
Explanation:
To solve this system of equations:
x + 2z = 4
x + y + z = 6
3x + 3y + 4z = 28
You can use the method of substitution or elimination. Let's use elimination:
First, we can simplify equation (3) by dividing both sides by 3 to make the coefficients of x and y match those in equations (1) and (2):
3x + 3y + 4z = 28 becomes x + y + (4/3)z = 28/3
Now, the system of equations looks like this:
x + 2z = 4
x + y + z = 6
x + y + (4/3)z = 28/3
Subtract equation (2) from equation (3) to eliminate y:
(x + y + (4/3)z) - (x + y + z) = 28/3 - 6
Now, simplify:
(4/3)z - z = 28/3 - 6
(4/3 - 1)z = 28/3 - 18/3
(1/3)z = 10/3
Now, solve for z:
z = (10/3) / (1/3) = 10
Now that we have the value of z, we can substitute it back into equation (2) to find y:
x + y + 10 = 6
y + x = -4
Now, we have two equations:
x + 2z = 4
y + x = -4
You can solve these equations simultaneously. Let's use equation (2) to express y in terms of x:
y = -4 - x
Now, substitute this into equation (1):
x + 2z = 4
x + 2(10) = 4
x + 20 = 4
x = 4 - 20
x = -16
So, the solution to the system of equations is:
x = -16
y = -4 - x = -4 - (-16) = 12
z = 10