Could you please show me how to do this?
It says to use the elimination method.
2x-y+2z=-7
-x+2y-4z=5
x+4y-6z=-1
http://library.thinkquest.org/C0110248/algebra/simelimination.htm
I know how to use the elimination method, I just get really confused when there's three lines and three variables.
Certainly! To solve the given system of equations using the elimination method, follow these steps:
Step 1: Start by arranging the equations in standard form, where the variables are in the same order.
2x - y + 2z = -7 ---> Equation 1
-x + 2y - 4z = 5 ---> Equation 2
x + 4y - 6z = -1 ---> Equation 3
Step 2: Pick two equations and eliminate one variable. Let's eliminate the variable "x" between equations 2 and 3.
Multiply Equation 2 by 2:
-2x + 4y - 8z = 10 ---> Equation 4
Add Equation 4 and Equation 3:
(-x + 2y - 4z) + (x + 4y - 6z) = 5 + (-1)
This will eliminate the "x" variable.
Simplifying this equation, we get:
6y - 10z = 4 ---> Equation 5
Step 3: Now, pick another pair of equations to eliminate a different variable. Let's eliminate the "x" variable between equations 1 and 3.
Multiply Equation 3 by 2:
2x + 8y - 12z = -2 ---> Equation 6
Add Equation 1 and Equation 6:
(2x - y + 2z) + (2x + 8y - 12z) = -7 + (-2)
This will eliminate the "x" variable.
Simplifying this equation, we get:
7y - 10z = -9 ---> Equation 7
Step 4: Now, we have two equations with two variables:
6y - 10z = 4 ---> Equation 5
7y - 10z = -9 ---> Equation 7
We can now eliminate either "y" or "z" to solve for the remaining variable.
To eliminate the "y" variable, multiply Equation 5 by 7 and Equation 7 by 6:
42y - 70z = 28 ---> Equation 8
42y - 60z = -54 ---> Equation 9
Subtract Equation 9 from Equation 8:
(42y - 70z) - (42y - 60z) = 28 - (-54)
Simplifying this equation, we get:
10z = 82
Step 5: Solve for "z" by dividing both sides of Equation 10 by 10:
z = 82/10
z = 8.2
Step 6: Substitute the found value of "z" back into either Equation 5 or 7 to solve for "y." Let's use Equation 5:
6y - 10(8.2) = 4
6y - 82 = 4
6y = 86
y = 86/6
y = 14.33 (rounded to two decimal places)
Step 7: Substitute the values of "y" and "z" back into any of the original equations to solve for "x." Let's use Equation 1:
2x - (14.33) + 2(8.2) = -7
2x - 14.33 + 16.4 = -7
2x + 2.07 = -7
2x = -7 - 2.07
2x = -9.07
x = -9.07/2
x = -4.54 (rounded to two decimal places)
So, the solution to the system of equations is:
x = -4.54, y = 14.33, z = 8.2