Step-by-Step Elimination Procedure
The elimination method for solving systems of linear equations follows a four-step procedure in enVision Algebra 1 Chapter 4 for Grade 11. First write both equations in standard form Ax + By = C. Second, multiply one or both equations so that the coefficients of one variable become opposites. Third, add the equations to eliminate that variable and solve the resulting single-variable equation. Fourth, substitute back to find the other variable. For example, solving {4x + 3y = 1, x - 3y = 4}: the y-terms are already opposites, so adding gives 5x = 5, x = 1, then substituting gives y = -1, solution (1, -1).
Key Concepts
Step 1. Write both equations in standard form $Ax + By = C$. Step 2. Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. Step 3. Add the equations resulting from Step 2 to eliminate one variable. Step 4. Solve for the remaining variable. Step 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. Step 6. Write the solution as an ordered pair. Step 7. Check that the ordered pair is a solution to both original equations.
Common Questions
What are the four steps of the elimination method?
1) Write both equations in standard form Ax + By = C. 2) Multiply to make one variable's coefficients opposites. 3) Add the equations to eliminate that variable and solve. 4) Substitute the found value back to solve for the other variable.
How do you solve {4x + 3y = 1, x - 3y = 4} by elimination?
The y-coefficients are already opposites (3 and -3). Add the equations: 5x = 5, so x = 1. Substitute into x - 3y = 4: 1 - 3y = 4, so y = -1. Solution: (1, -1).
When do you need to multiply before adding?
When no variable has opposite coefficients yet. For {2x + y = 5, x + y = 3}, multiply the second equation by -1 to get -x - y = -3, then add to eliminate y.
Can you choose which variable to eliminate?
Yes. Choose the variable whose coefficients are easiest to make opposite with minimal multiplication. If one variable already has opposite coefficients, eliminate it immediately.
What do you do after solving for one variable?
Substitute the value back into either original equation and solve for the second variable. Then verify your solution by checking it in both original equations.