Final answer:
To eliminate variables using the elimination method, align the equations to cancel out one variable by adding or subtracting them, then solve the resulting new equations for the remaining variables. Finally, check the answers by substituting them back into the original equations.
Step-by-step explanation:
To solve the system of equations using the elimination method, you start by aligning the equations so you can eliminate variables. Looking at your system of equations:
-2x + 5y - 2z = 6
7x - 7y + 2z = -9
7x - 3y + 2z = 7
You can eliminate the variable z by adding the second and third equations, because 2z and 2z will neutralize each other.
7x - 7y + 2z = -9
+ 7x - 3y + 2z = 7
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14x - 10y = -2
Now you have a new simplified equation 14x - 10y = -2. Next, find another pair of equations where you can eliminate the same or different variable. For instance, if you multiply the first equation by 7 and add it to the second, z will again be eliminated:
-14x + 35y - 14z = 42
+ 7x - 7y + 2z = -9
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-7x + 28y - 12z = 33
With these steps, you have two equations with x and y only. Finally, solve these two new equations for the variables x and y and then substitute the found values into any of the original equations to solve for z. Remember to check your answers by substituting them back into the original equations to ensure they satisfy all three equations.