Elimination Method

The elimination method (or method of elimination by equating coefficients) is a technique for solving a pair of linear equations by manipulating the equations so that the addition or subtraction of the two equations eliminates one of the variables.

Core idea: If we can make the coefficients of one variable (say y) equal in both equations but with opposite signs, then adding the equations will eliminate y. If the coefficients are already equal with the same sign, subtracting the equations will eliminate y.

The procedure involves:

1. Multiplying one or both equations by suitable non-zero constants so that the coefficients of one variable become equal in magnitude.

2. Adding or subtracting the equations to eliminate that variable.

3. Solving the resulting single-variable equation.

4. Substituting back to find the other variable.

What can happen during elimination:

• If one variable is eliminated and a valid equation in the other variable remains, the system has a unique solution.
• If both variables are eliminated and the result is a true statement (like 0 = 0), the system has infinitely many solutions.
• If both variables are eliminated and the result is a false statement (like 0 = 7), the system has no solution.

The elimination method is justified by the properties of equality and the ability to perform the same operation on both sides of an equation.

Principle 1: Scaling invariance. If a1*x + b1*y = c1 is true, then multiplying both sides by any non-zero constant k gives k*a1*x + k*b1*y = k*c1, which is also true. The solution set is unchanged.

Principle 2: Addition property. If equation (i) holds and equation (ii) holds, then their sum (i) + (ii) also holds for the same values of x and y. Similarly, (i) - (ii) holds. This is because equals added to equals give equals.

Combining these principles:

Given: a1*x + b1*y = c1 ... (i) and a2*x + b2*y = c2 ... (ii).

Multiply (i) by b2 and (ii) by b1:

a1*b2*x + b1*b2*y = c1*b2 ... (iii)

a2*b1*x + b1*b2*y = c2*b1 ... (iv)

Subtract (iv) from (iii): (a1*b2 - a2*b1)*x = c1*b2 - c2*b1.

The y terms cancel because both have coefficient b1*b2*y. This is the essence of elimination — by careful choice of multipliers, we engineer the cancellation of one variable.

When a1*b2 - a2*b1 = 0: This means a1/a2 = b1/b2 (the lines have equal slopes). The elimination of one variable also eliminates the other, leaving either 0 = 0 (infinitely many solutions) or 0 = nonzero (no solution).

Connection to determinants: The expression a1*b2 - a2*b1 is the determinant of the coefficient matrix. In linear algebra, this is denoted as det(A) = |a1 b1; a2 b2| = a1*b2 - a2*b1. The system has a unique solution if and only if this determinant is non-zero. This connection foreshadows matrix methods studied in higher mathematics.

Different types of elimination scenarios arise based on the coefficients:

Type 1: Coefficients already equal (opposite signs)

Example: 3x + 2y = 11 and 3x - 2y = 5. The y coefficients are +2 and -2. Simply add the equations: 6x = 16, x = 8/3. This is the easiest case.

Type 2: Coefficients already equal (same sign)

Example: 5x + 3y = 11 and 2x + 3y = 5. The y coefficients are both +3. Subtract the equations: 3x = 6, x = 2.

Type 3: One coefficient is a multiple of the other

Example: 2x + 3y = 7 and 4x + y = 13. Multiply the first equation by 2: 4x + 6y = 14. Now subtract from 4x + y = 13 (or vice versa): (4x + 6y) - (4x + y) = 14 - 13, giving 5y = 1.

Type 4: No simple multiple — multiply both equations

Example: 3x + 4y = 10 and 5x + 6y = 16. To eliminate x, multiply first by 5 and second by 3: 15x + 20y = 50 and 15x + 18y = 48. Subtract: 2y = 2, y = 1.

Type 5: Equations need simplification first

Example: 0.2x + 0.3y = 1.3 and 0.4x + 0.5y = 2.3. Multiply by 10 to clear decimals: 2x + 3y = 13 and 4x + 5y = 23. Then apply standard elimination.

Step-by-Step Procedure for the Elimination Method:

Step 1: Write both equations in standard form. Make sure both equations are in the form ax + by = c with like terms aligned.

Step 2: Choose which variable to eliminate. Look at the coefficients and decide which variable is easier to eliminate. If one variable already has equal (or opposite) coefficients, choose that one.

Step 3: Make the coefficients equal. Multiply one or both equations by appropriate constants so that the coefficients of the chosen variable have the same absolute value.

To eliminate x: Make coefficients of x equal. Multiply equation (i) by |a2| and equation (ii) by |a1| (or use the LCM of a1 and a2).

To eliminate y: Make coefficients of y equal. Multiply equation (i) by |b2| and equation (ii) by |b1| (or use the LCM of b1 and b2).

Step 4: Add or subtract the equations. If the equal coefficients have opposite signs, add the equations. If they have the same sign, subtract one from the other.

Step 5: Solve the resulting equation. After elimination, you have a single equation in one variable. Solve it.

Step 6: Substitute back. Plug the value from Step 5 into either original equation to find the other variable.

Step 7: Verify. Substitute both values into both original equations to confirm.

Efficient choices:

• If one coefficient is already 1, multiply only the other equation.
• Use the LCM of the coefficients (rather than the product) to keep numbers small.
• When coefficients are already opposites, just add — no multiplication needed.

The elimination method is used extensively in mathematics, science, and practical problem-solving.

Engineering: Electrical circuits with multiple loops lead to systems of equations (Kirchhoff's laws). The elimination method is used to solve for unknown currents and voltages in these circuits.

Business and Finance: Problems involving two types of investments, two products, or two cost components are modelled as pairs of linear equations and solved using elimination to find the individual values.

Chemistry: Balancing complex chemical equations sometimes requires solving systems of linear equations. The elimination method helps determine the coefficients of reactants and products.

Sports Analytics: When two players' statistics satisfy certain total and difference conditions, the elimination method can be used to find individual statistics.

Traffic Flow: In urban planning, traffic at intersections is modelled using systems of equations. The flow rates on different roads can be determined using elimination.

Linear Algebra: The elimination method is the foundation of Gaussian elimination, a systematic procedure used to solve systems of any number of equations with any number of unknowns. This is one of the most important algorithms in computational mathematics.