Solve Systems of Three Equations by Elimination

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When you have a system of 3 equations, you have 3 variables that you need to solve for. The process of solving for these variables is similar to that of solving for 2 variables using elimination.

Here are the steps to follow:

Write the 3 equations in standard form (with the variables on the left side and constants on the right side).

Choose 2 equations and eliminate one variable. You can do this by adding or subtracting the equations in such a way that one variable is eliminated.

Write the resulting equation with the 2 remaining variables.

Choose 2 other equations and eliminate the same variable you just eliminated in step 2.

Write the resulting equation with the 2 remaining variables.

You should now have 2 equations with 2 variables. Solve for one variable using either substitution or elimination.

Substitute the value of the solved variable into any of the original equations to find the values of the other 2 variables.

Check your solution by substituting the values you found into all 3 equations.

Here's an example:

x + y + z = 6 2x - y + 3z = 2 x + y - z = 0

We can start by eliminating z from the first 2 equations. To do this, we can add the first equation to 3 times the second equation:

x + y + z = 6 6x - 3y + 9z = 18 7x - 2y = 24

Next, we can eliminate z from the first and third equations. To do this, we can subtract the third equation from the first:

x + y + z = 6 x + y - z = 0 2x + 2y = 6

Now we have 2 equations with 2 variables, which we can solve using elimination:

7x - 2y = 24 2x + 2y = 6 9x = 30

x = 30/9 = 10/3

We can substitute this value of x into one of the previous equations to solve for y:

2(10/3) + 2y = 6

4/3 + 2y = 6

2y = 17/3

y = 17/6

Finally, we can substitute these values of x and y into one of the previous equations to solve for z:

10/3 + 17/6 + z = 6

z = 5/6

Therefore, the solution to the system of equations is:

x = 10/3, y = 17/6, z = 5/6

Remember to check your solution by substituting the values you found into all 3 equations.

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