1. Introduction
The elimination method is a systematic way to solve a pair of linear equations by eliminating one variable. This is done by adding or subtracting the equations after making the coefficients of one variable equal. This method is widely used because it works for all types of linear equations in two variables.
2. Steps of the Elimination Method
To solve two equations using the elimination method, follow these steps:
2.1. Step 1: Arrange the Equations
Write both equations in the standard form:
\(a_1x + b_1y = c_1\)
\(a_2x + b_2y = c_2\)
2.2. Step 2: Make the Coefficients Equal
Multiply one or both equations so that the coefficients of either \(x\) or \(y\) become equal in magnitude.
Example:
To eliminate \(x\): If equations are \(2x + 3y = 11\) and \(3x - y = 4\), multiply the first by 3 and the second by 2.
2.3. Step 3: Add or Subtract the Equations
Once the coefficients match, add or subtract the equations to eliminate one variable.
Example:
\(6x + 9y = 33\)
\(6x - 2y = 8\)
Subtract to eliminate \(x\):
\(11y = 25\)
2.4. Step 4: Solve the Resulting Equation
The new equation has only one variable. Solve it normally.
2.5. Step 5: Substitute Back
Use the found value to substitute in any original equation to find the other variable.
2.6. Step 6: Write Final Answer
Write the final solution as an ordered pair \((x, y)\).
3. Worked Examples
Let’s understand the elimination method through examples.
3.1. Example 1: Simple Coefficients
Solve:
\(3x + 4y = 10\)
\(3x - 2y = 4\)
Step 1: Coefficients of \(x\) are already equal.
Step 2: Subtract the second equation from the first:
\((3x + 4y) - (3x - 2y) = 10 - 4\)
\(6y = 6\)
\(y = 1\)
Step 3: Substitute \(y = 1\) in \(3x + 4y = 10\)
\(3x + 4 = 10\)
\(3x = 6\)
\(x = 2\)
Solution: \((2, 1)\)
3.2. Example 2: Scaling Required
Solve:
\(2x + 3y = 13\)
\(3x + 5y = 19\)
Step 1: To eliminate \(x\), multiply first equation by 3 and second by 2:
\(6x + 9y = 39\)
\(6x + 10y = 38\)
Step 2: Subtract:
\(y = -1\)
Step 3: Substitute into first equation:
\(2x + 3(-1) = 13\)
\(2x - 3 = 13\)
\(2x = 16\)
\(x = 8\)
Solution: \((8, -1)\)
4. Special Cases
The elimination method reveals special cases based on the resulting equation.
4.1. Case 1: No Solution (Parallel Lines)
If elimination gives an equation like:
\(0x + 0y = 5\)
This is false → the lines are parallel → no solution.
4.2. Case 2: Infinite Solutions (Coincident Lines)
If elimination gives:
\(0x + 0y = 0\)
This is always true → both equations represent the same line → infinitely many solutions.
5. Common Mistakes
- Multiplying incorrectly while matching coefficients.
- Sign errors during addition or subtraction.
- Substituting back into the wrong equation.
- Not reducing the final equation properly.
6. Quick Practice
Solve using the elimination method:
- \(x + y = 9\), \(x - y = 3\)
- \(2x - 3y = 5\), \(3x + y = 8\)
- \(4a + 6b = 18\), \(2a - 3b = -6\)
7. Summary
- Elimination method removes one variable by adding or subtracting equations.
- Make coefficients equal before eliminating.
- After elimination, solve the resulting single-variable equation.
- Always check for special cases: no solution or infinite solutions.