Solve the system by substitution.
-x-y-z=-8
-4x+4y+5z=7
2x+2z=4
I am very confused about this. Can someone please explain how to do this?
I am super confused.
Darn I really hope Reiny comes on.
the name substitution tells you how to solve the system.
From the first equation, you know that
z = 8-x-y
Now, substitute that into the other two equations. That gives you
-4x+4y+5(8-x-y)=7
2x+2(8-x-y)=4
or,
-9x-y = -33
-2y = -12
Now you can see that
y=6
Thus, -9x-6=-33, so x = 3
And finally, z=8-3-6 = -1
To solve this system by substitution, we need to first solve one equation for one variable and then substitute that expression into the other equations. Let's follow these steps to solve this system:
Step 1: Solve one equation for one variable.
Let's solve the third equation, 2x + 2z = 4, for x:
2x = 4 - 2z
x = (4 - 2z) / 2
x = 2 - z
Step 2: Substitute the expression for x into the other equations.
Now that we have solved the third equation for x, we can substitute the expression (2 - z) for x in the first and second equations:
1st Equation: -x - y - z = -8
Substitute x = 2 - z:
-(2 - z) - y - z = -8
-2 + z - y - z = -8
-2 - 2z - y = -8
-2 - y - 2z = -8
-y - 2z = -8 + 2
-y - 2z = -6
y + 2z = 6
This is our new 1st equation.
2nd Equation: -4x + 4y + 5z = 7
Substitute x = 2 - z:
-4(2 - z) + 4y + 5z = 7
-8 + 4z + 4y + 5z = 7
-8 + 4y + 9z = 7
4y + 9z = 7 + 8
4y + 9z = 15
This is our new 2nd equation.
Now we have a system of two equations with two variables:
y + 2z = 6 (Equation 1)
4y + 9z = 15 (Equation 2)
Step 3: Solve the new system.
To solve this system, we can use any method we prefer, such as substitution or elimination:
In this case, let's solve the first equation for y and substitute it into the second equation:
Equation 1: y + 2z = 6
Solve for y:
y = 6 - 2z
Substitute y = 6 - 2z into Equation 2:
4(6 - 2z) + 9z = 15
24 - 8z + 9z = 15
24 + z = 15
z = 15 - 24
z = -9
Now substitute z = -9 into Equation 1 to solve for y:
y + 2(-9) = 6
y - 18 = 6
y = 6 + 18
y = 24
Finally, substitute the values of y and z into the expression for x that we found earlier:
x = 2 - z
x = 2 - (-9)
x = 2 + 9
x = 11
Therefore, the solution to the given system of equations is:
x = 11, y = 24, z = -9.