solve the system by substitution.
-x-y-z=-8
-4x+4y+5z=7
2x+2z=4
please help ;-;
2x+2z=4
is
z = 2 - x
then
-x-y-z=-8
is
- x - y -2 + x = -8
or
-y = -6
which is y = 6
-4x+4y+5z=7
is
-4x + 24 +5(2-x) = 7
-4 x + 34 -5 x = 7
- 9 x = -27
x = 3
now go back and get z
To solve the system of equations by substitution, we'll solve one of the equations for one variable and substitute that expression into the other equations.
Let's solve the third equation for x:
2x + 2z = 4
2x = 4 - 2z
x = 2 - z
Now, we can substitute the expression for x in the first and second equations:
Equation 1: -x - y - z = -8
-(2 - z) - y - z = -8
-2 + z - y - z = -8
-z - y = -6 (equation A)
Equation 2: -4x + 4y + 5z = 7
-4(2 - z) + 4y + 5z = 7
-8 + 4z + 4y + 5z = 7
9z + 4y = 15 (equation B)
Now we have a system of two equations:
Equation A: -z - y = -6
Equation B: 9z + 4y = 15
From Equation A, let's solve it for y:
-y = -6 + z
y = 6 - z
We can substitute this expression for y into Equation B:
9z + 4(6 - z) = 15
9z + 24 - 4z = 15
5z = -9
z = -9/5
Now we can substitute z = -9/5 into either Equation A or B. Let's use Equation A:
(-1)(-9/5) - y = -6
9/5 - y = -6
y = 9/5 + 6
y = 9/5 + 30/5
y = 39/5
Finally, substituting the values of y and z into the expression for x:
x = 2 - z
x = 2 - (-9/5)
x = 2 + 9/5
x = 10/5 + 9/5
x = 19/5
Therefore, the solution to the system of equations is:
x = 19/5
y = 39/5
z = -9/5