Systems of Equations (3+ variables)

Definition: To solve a system of equations with three or more variables, the same methods as solving a system of equations with two variables will be helpful. We can still use substitution and elimination (see: Solving Systems of Linear Equations).

Example:

Solve for x, y, and z:

Solution:

Step 1: Using the elimination method, add the first two equations to cancel out the z:

Step 2: Multiply the top equation by 2. This will allow us to eliminate the z when adding the top and bottom equations:

Step 3: Working with this newest equation, divide both sides by 7, then isolate the x:

Step 4: Now that we know x = (7 – y), substitute this into the equation found in Step 1 and simplify to solve for y:

Step 5: Substitute y = 5 into the equation found in Step 3 to solve for x:

Step 6: Plug x = 2 and y = 5 into any of the 3 equations to solve for z:

Our answer will be x = 2, y = 5, z = 4.

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