Tenth Grade Mathematics: Algebraic Solution: Method of Elimination

While the substitution method works well when one variable has a unit coefficient, it often introduces cumbersome fractional arithmetic when all coefficients are integers greater than one. The Method of Elimination—specifically, elimination by equating coefficients—provides a streamlined algebraic alternative. By multiplying one or both equations by suitable non-zero constants, we match the coefficients of one variable in magnitude, allowing us to add or subtract the equations to eliminate that variable completely in a single stroke.

The Principle of Equating Coefficients

The elimination method relies on the fundamental additive property of equality: if A = B and C = D, then A + C = B + D, and A - C = B - D.

Step-by-Step Elimination Protocol:
Step 1: Write both equations in standard form: a*x + b*y = c.
Step 2: Choose one variable to eliminate (either x or y).
Step 3: Find the Lowest Common Multiple (LCM) of the absolute values of the
        coefficients of the chosen variable.
Step 4: Multiply one or both equations by appropriate non-zero integers so that
        the coefficients of the chosen variable become numerically equal.
Step 5: If the matching coefficients have OPPOSITE signs, ADD the two equations.
        If the matching coefficients have the SAME sign, SUBTRACT one equation
        from the other.
Step 6: Solve the resulting single-variable equation.
Step 7: Substitute the solved value into any original equation to determine the
        other variable.

Sign Rule Summary for Step 5:
   Coefficients have Same Sign (+ and +, or - and -)     ===> SUBTRACT
   Coefficients have Opposite Signs (+ and -, or - and +) ===> ADD

Detailed Worked Examples

Example 1: Eliminating Variable with Opposite Signs

Solve the linear system by elimination:

Equation 1:  9*x - 4*y = 2000
Equation 2:  7*x - 3*y = 2000

Step 1: Choose to eliminate y. The coefficients of y are -4 and -3. The LCM of 4 and 3 is 12. Step 2: Multiply Equation 1 by 3, and multiply Equation 2 by 4:

3 * (9*x - 4*y) = 3 * 2000  ===>  27*x - 12*y = 6000    --- (Equation 3)
4 * (7*x - 3*y) = 4 * 2000  ===>  28*x - 12*y = 8000    --- (Equation 4)

Step 3: The coefficients of y are identical in sign (-12 and -12), so we subtract Equation 3 from Equation 4:

  (28*x - 12*y) - (27*x - 12*y) = 8000 - 6000
  28*x - 27*x - 12*y + 12*y = 2000
  x = 2000

Step 4: Substitute x = 2000 into Equation 1 to find y:

9*(2000) - 4*y = 2000
18000 - 4*y = 2000
-4*y = 2000 - 18000
-4*y = -16000
y = 4000

The unique solution is x = 2000, y = 4000.

Symmetric Coefficient Technique

An elegant shortcut arises in systems where the coefficients of x and y are interchanged between the two equations:

Special Form:
a*x + b*y = c1
b*x + a*y = c2

Instead of multiplying by large numbers to equate coefficients, execute the two-step addition-subtraction method: Add the two equations: (a + b)x + (a + b)y = c1 + c2 ===> x + y = (c1 + c2) / (a + b). Subtract the two equations: (a - b)x - (a - b)y = c1 - c2 ===> x - y = (c1 - c2) / (a - b). Solve the resulting elementary system of x + y and x - y by direct addition.

Worked Demonstration:
Equation 1:  37*x + 43*y = 123
Equation 2:  43*x + 37*y = 117

Step 1: Add Equations 1 and 2:
        80*x + 80*y = 240  ===>  Divide by 80:  x + y = 3     --- (Equation A)

Step 2: Subtract Equation 1 from Equation 2:
        6*x - 6*y = -6     ===>  Divide by 6:   x - y = -1    --- (Equation B)

Step 3: Add Equations A and B:
        (x + y) + (x - y) = 3 + (-1)  ===>  2*x = 2  ===>  x = 1.
        Then y = 3 - 1 = 2.
Solution: x = 1, y = 2.

Chapter Exercises

Exercise 1: Solve the pair of linear equations by the elimination method: 3x + 4y = 10 and 2x - 2y = 2.

Exercise 2: Solve by elimination: (x/2) + (2*y/3) = -1 and x - (y/3) = 3.

Exercise 3: Solve the symmetric-coefficient system using the addition-subtraction shortcut: 152x - 378y = -74 and -378x + 152y = -604.

Solutions and Explanations

Solution 1: We have 3x + 4y = 10 (Equation 1) and 2x - 2y = 2 (Equation 2). To eliminate y, multiply Equation 2 by 2: 2(2x - 2y) = 2(2) ==> 4x - 4y = 4 (Equation 3). The coefficients of y in Equation 1 and Equation 3 are +4 and -4 (opposite signs). Add Equation 1 and Equation 3: (3x + 4y) + (4x - 4y) = 10 + 4 ==> 7x = 14 ==> x = 2. Substitute x = 2 into Equation 1: 3(2) + 4y = 10 ==> 6 + 4y = 10 ==> 4*y = 4 ==> y = 1. The solution is x = 2, y = 1.

Solution 2: First clear denominators. Multiply (x/2) + (2y/3) = -1 by 6: 3x + 4y = -6 (Equation 1). Multiply x - (y/3) = 3 by 3: 3x - y = 9 (Equation 2). Notice that both equations have identical x-coefficients (+3). Subtract Equation 2 from Equation 1: (3x + 4y) - (3x - y) = -6 - 9 ==> 5y = -15 ==> y = -3. Substitute y = -3 into Equation 2: 3x - (-3) = 9 ==> 3x + 3 = 9 ==> 3*x = 6 ==> x = 2. Thus, the solution is x = 2, y = -3.

Solution 3: We use the symmetric addition-subtraction method. Equation 1: 152x - 378y = -74. Equation 2: -378x + 152y = -604. Adding Equation 1 and Equation 2 gives: (152 - 378)x + (-378 + 152)y = -74 - 604 ==> -226x - 226y = -678. Dividing both sides by -226 yields: x + y = 3 (Equation A). Now subtract Equation 2 from Equation 1: [152 - (-378)]x + [-378 - 152]y = -74 - (-604) ==> 530x - 530y = 530. Dividing both sides by 530 yields: x - y = 1 (Equation B). Adding Equation A and Equation B: 2*x = 4 ==> x = 2. Substituting x = 2 into Equation A gives: 2 + y = 3 ==> y = 1. Thus, the solution is x = 2, y = 1.