Can someone please help me I'd really appreciate it
Solve the system by substitution
-x - y - z = -8
-4x + 4y + 5z = 7
2x + 2z =4
Certainly! I'd be happy to help you solve the system of equations using the substitution method.
Step 1: Solve one equation for one variable in terms of the other variable.
Let's start with the third equation, 2x + 2z = 4. We can solve this equation for x by isolating x:
2x = 4 - 2z
x = (4 - 2z) / 2
x = 2 - z
Now, we have x expressed in terms of z.
Step 2: Substitute the expression we found for x into the other two equations of the system.
Substitute x = 2 - z into the first equation:
-x - y - z = -8
-(2 - z) - y - z = -8
-2 + z - y - z = -8
-2 - y = -8 + 2z
Now substitute x = 2 - z into the second equation:
-4x + 4y + 5z = 7
-4(2 - z) + 4y + 5z = 7
-8 + 4z + 4y + 5z = 7
-8 + 9z + 4y = 7
Step 3: We now have a system of two equations with two variables:
-2 - y = -8 + 2z (Equation 1)
-8 + 9z + 4y = 7 (Equation 2)
Step 4: Solve for one variable in terms of the other and substitute it back into the other equation.
From Equation 1, we can solve for y:
y = -8 + 2z + 2
Substitute y = -8 + 2z + 2 into Equation 2:
-8 + 9z + 4(-8 + 2z + 2) = 7
-8 + 9z - 32 + 8z + 8 = 7
17z - 32 = 7
17z = 7 + 32
17z = 39
z = 39 / 17
z = 2.29 (rounded to two decimal places)
Step 5: Substitute the value of z back into the expression for y:
y = -8 + 2z + 2
y = -8 + 2(2.29) + 2
y = -8 + 4.58 + 2
y = -1.42 (rounded to two decimal places)
Step 6: Substitute the values of y and z back into the expression for x:
x = 2 - z
x = 2 - 2.29
x = -0.29 (rounded to two decimal places)
Therefore, the solution to the given system of equations is:
x ≈ -0.29, y ≈ -1.42, z ≈ 2.29.
I hope this helps! Let me know if you have any other questions.