The Elimination Method - Algebra-1

1. Fundamental Concepts

Definition: The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to eliminate one of the variables.
Objective: To find the values of the variables that satisfy all equations in the system simultaneously.
Steps:
1. Multiply one or both equations by a constant to make the coefficients of one variable opposites.
2. Add or subtract the equations to eliminate one variable.
3. Solve for the remaining variable.
4. Substitute the value back into one of the original equations to find the other variable.

Goal: Simplify the system by eliminating one variable, and ultimately find the solution to the system (i.e., the values of the unknowns that satisfy both equations).
Applicable Conditions: If the coefficients of a variable in the two equations are equal or opposite, you can directly add or subtract to eliminate the variable; if not, you need to multiply by an appropriate constant first to make the coefficients of a variable equal or opposite.
Solution Scenarios:
- If an always true equation (e.g., ) is obtained after elimination, the system has infinitely many solutions.
- If an always false equation (e.g., ) is obtained, the system has no solution.
- If a single-variable linear equation (e.g., ) is obtained, the system has a unique solution.

Easy Level Solve the system:
Solution:
The coefficients of y in the two equations are opposites (1 and -1). Add the two equations to eliminate y: Simplify: , so .
Substitute into the first equation: , so . The solution to the system is .
Medium Level Solve the system:
Solution:
Observe the coefficients: the coefficients of x are 2 and 4 (in a multiple relationship). Multiply the first equation by 2 to make the coefficients of x equal: First equation × 2: (denoted as Equation ③)
Subtract the second equation from Equation ③ to eliminate x: Simplify: , so .
Substitute into the second equation: , so . The solution to the system is .
Hard Level
Solve the system:
Solution: To eliminate y, multiply the equations by constants to make the coefficients of y opposites: First equation × 3: (denoted as Equation ③) Second equation × 2: (denoted as Equation ④)
Add Equation ③ and Equation ④ to eliminate y: Simplify: , so .
Substitute into the first equation: , so . The solution to the system is .

Observe Coefficients: Prioritize eliminating variables with smaller absolute coefficients or those in a multiple relationship to reduce calculations.
Adjust Coefficients: If coefficients of a variable are neither equal nor opposite, multiply by divisors of the least common multiple of the two coefficients to make them equal or opposite.
Verify Solutions: Substitute the obtained solution back into both original equations to check if they satisfy the equalities.
Identify Special Cases: If \(0 = 0\) (infinitely many solutions) or \(0=\text{non-zero number}\) (no solution) appears after elimination, directly determine the solution scenario of the system without further calculation.