Solve the system by substitution.
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⎪
⎨
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−
x
−
y
−
z
=
−
8
−
4
x
+
4
y
+
5
z
=
7
2
x
+
2
z
=
4
From the third equation, we can solve for x in terms of z:
2x + 2z = 4
2x = -2z + 4
x = -z + 2
We can substitute this expression for x into the second equation:
-4(-z + 2) + 4y + 5z = 7
4z - 8 + 4y + 5z = 7
9z + 4y = 15
Now we have two equations and two variables:
x = -z + 2
9z + 4y = 15
We can solve for one variable in terms of the other and substitute back:
4y = 15 - 9z
y = (15 - 9z)/4
x = -z + 2
Substitute into the first equation:
-x - y - z = -8
-(-z + 2) - (15 - 9z)/4 - z = -8
z/2 - 7/2 = -8
z/2 = -1/2
z = -1
Now we can solve for y and x:
y = (15 - 9(-1))/4 = 6
x = -(-1) + 2 = 3
Therefore, the solution is (x, y, z) = (3, 6, -1).
To solve the system of equations by substitution, we can start by solving one of the equations for one variable and then substituting it into the remaining equations.
Let's solve the third equation, 2x + 2z = 4, for x:
2x = 4 - 2z
x = (4 - 2z)/2
x = 2 - z
Now we can substitute this expression for x into the other equations in the system:
Substituting x = 2 - z into the first equation, we have:
-(2 - z) - y - z = -8
-2 + z - y - z = -8
-z - y - 2 = -8
-z - y = -6 --- (Equation 1)
Substituting x = 2 - z into the second equation, we have:
-4(2 - z) + 4y + 5z = 7
-8 + 4z + 4y + 5z = 7
9z + 4y = 15 --- (Equation 2)
We now have a system of two equations in the variables y and z:
Equation 1: -z - y = -6
Equation 2: 9z + 4y = 15
Now we can solve this system of equations by using the method of substitution.
From Equation 1, we can solve for y:
y = -6 - z
Substituting y = -6 - z into Equation 2:
9z + 4(-6 - z) = 15
9z - 24 - 4z = 15
5z - 24 = 15
5z = 15 + 24
5z = 39
z = 39/5
z = 7.8
Now substituting z = 7.8 into Equation 1:
-7.8 - y = -6
y = -6 + 7.8
y = 1.8
Finally, we can substitute the values of y = 1.8 and z = 7.8 back into our expression for x = 2 - z:
x = 2 - 7.8
x = -5.8
Therefore, the solution to the system of equations is:
x = -5.8, y = 1.8, and z = 7.8.