Solving by Elimination (Addition/Subtraction Method)

TL;DR

To solve a 3x3 system by elimination, you'll strategically add or subtract equations to remove one variable, creating a 2x2 system. You then solve the 2x2 system and substitute the values back into one of the original equations to find the last variable. This method is great when variables have matching or opposite coefficients.

1. The Mental Model

Think of this method like a detective narrowing down suspects. You have three clues (equations) with three unknowns (variables). You combine clues in pairs to get rid of one unknown, leaving you with simpler clues (fewer equations, fewer unknowns) until you can identify one unknown directly.

2. The Core Material

When solving a 3x3 system of equations using elimination, your goal is to reduce the system down to a 2x2 system, and then eventually to a 1x1 system (a single equation with one variable).

Step 1: Choose a Variable to Eliminate

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Look at your three equations. Pick one variable (x, y, or z) that you want to eliminate first. It's often easiest if two equations already have the same or opposite coefficients for that variable. If not, you'll need to multiply one or both equations by a constant to make them match.

Step 2: Create Two New Equations (a 2x2 System)

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You'll pair up the original equations in two different ways to eliminate the same variable you chose in Step 1.

1. Pair 1: Combine two of the original equations to eliminate your chosen variable. This will give you a new equation with only two variables.
2. Pair 2: Combine a different pair of the original equations (make sure to use the third original equation here) to eliminate the same variable. This will give you a second new equation with only two variables.

Now you have a system of two equations with two variables.

Step 3: Solve the 2x2 System

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Use elimination (or substitution, if you prefer) on your new 2x2 system to solve for one of the remaining variables. Once you find that value, substitute it back into one of the 2x2 equations to find the second variable.

Step 4: Find the Third Variable

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Take the two variable values you just found and substitute them back into any of your original three equations. Solve for the last remaining variable.

Step 5: Check Your Solution

Substitute all three values (x, y, z) into all three original equations to ensure they are all true.

graph TD
    A["Start with 3 original equations (3 variables)"] --> B{Choose a variable to eliminate};
    B --> C["Pair Eq 1 & Eq 2, eliminate chosen variable"] --> D["New Eq A (2 variables)"];
    B --> E["Pair Eq 1 & Eq 3 (or Eq 2 & Eq 3), eliminate *same* chosen variable"] --> F["New Eq B (2 variables)"];
    D & F --> G["Solve the 2x2 system (New Eq A & New Eq B)"];
    G --> H["Find value for first variable"];
    H --> I["Substitute back into 2x2 system to find value for second variable"];
    I --> J["Substitute both found values into one original equation"];
    J --> K["Find value for third variable"];
    K --> L["Check all three values in all original equations"];
    L --> M["Solution found (x, y, z)"];

3. Worked Example

Let's solve the system:
1. x + y + z = 6
2. 2x - y + 3z = 9
3. -x + 2y + 2z = 9

Step 1: Choose a Variable to Eliminate
Let's eliminate y because equations (1) and (2) have +y and -y, which will cancel out easily.

Step 2: Create Two New Equations (a 2x2 System)

• Pair 1 (Eq 1 + Eq 2):
  (x + y + z = 6)
+ (2x - y + 3z = 9)
------------------
3x + 4z = 15 (This is our new Equation A)

• Pair 2 (Eq 1 and Eq 3):
  We need to make the y coefficients opposite. Multiply Eq 1 by -2:
  -2 * (x + y + z = 6) becomes -2x - 2y - 2z = -12
  Now add this to Eq 3:
  (-2x - 2y - 2z = -12)
+ (-x + 2y + 2z = 9)
------------------
-3x + 0y + 0z = -3
-3x = -3 (This is our new Equation B)

Step 3: Solve the 2x2 System
Our 2x2 system is:
A. 3x + 4z = 15
B. -3x = -3

From Equation B, we can directly solve for x:
-3x = -3
x = 1

Now substitute x = 1 into Equation A:
3(1) + 4z = 15
3 + 4z = 15
4z = 12
z = 3

Step 4: Find the Third Variable
We have x = 1 and z = 3. Substitute these into any original equation. Let's use Eq 1:
x + y + z = 6
1 + y + 3 = 6
4 + y = 6
y = 2

Step 5: Check Your Solution
Solution: x = 1, y = 2, z = 3
1. 1 + 2 + 3 = 6 (True)
2. 2(1) - 2 + 3(3) = 2 - 2 + 9 = 9 (True)
3. -1 + 2(2) + 2(3) = -1 + 4 + 6 = 9 (True)
All equations hold, so the solution is correct!

4. Key Takeaways

• Always aim to reduce a 3x3 system to a 2x2 system first.
• Choose one variable to eliminate consistently across two different pairs of equations.
• You might need to multiply one or both equations by constants to create matching or opposite coefficients.
• Carefully manage your signs when adding or subtracting equations.
• After solving the 2x2 system, substitute back to find the remaining variable.
• Always check your final solution by plugging the values into all original equations.

Common Mistakes to Avoid:
- Eliminating a different variable in your second pair of equations.
- Making arithmetic errors, especially with negative signs, when combining equations.
- Forgetting to multiply every term in an equation when scaling it.
- Incorrectly substituting values back into equations.
- Not checking your final answer, which can catch many errors.

5. Now Try It

Solve the following system using elimination. When you're done, your solution for x, y, and z should make all three equations true.

1. x - 2y + 3z = 7
2. 2x + y + z = 4
3. -3x + 2y - 2z = -10