Unit 2: Systems of Linear Equations and Inequalities
Topic 2
Solving Systems by Elimination
The elimination method is a way to solve a system of equations by adding or subtracting the equations so that one variable is removed (eliminated). This allows us to solve for the remaining variable.
For example:
If we add these two equations, is eliminated:
Then substitute back into one equation to find .
Core Skills
- Line up equations vertically by variables and equal signs.
- Decide whether to add or subtract to eliminate a variable.
- Multiply one or both equations if coefficients need adjustment.
- Always simplify and substitute back to check the solution.
Example 1: Direct Elimination
Solve the system:
Step 1: Add the two equations to eliminate :
Step 2: Substitute into :
Final Answer:
Example 2: Multiplying to Eliminate
Solve the system:
Step 1: Add the two equations to eliminate :
Step 2: Substitute into :
Final Answer:
Example 3: Multiplying Both Equations
Solve the system:
Step 1: Multiply the first equation by 3 and the second by 2 to make the -coefficients match:
Step 2: Subtract the second equation from the first:
Step 3: Substitute into :
Final Answer:
Key Takeaways
- Use elimination when variables are already aligned or easy to match.
- Multiply equations to create equal and opposite coefficients if needed.
- Always check the solution in both equations.