Decide whether the equations are more easily solved through substitution or elimination methods. Both approaches work; this decision is purely a matter of preference. In this example, elimination will be used.
Find a common multiple for the coefficient--numbers in front of--a variable. Say the equations are 3x - 7y = 8 (equation 1) and 4x + 5y = 11 (equation 2). The x-coefficients are 3 and 4; the y-coefficients are -7 and 5.
Pick which variable to eliminate and cross-multiply the equations by the coefficients for that variable. In the example given, 3x - 7y = 8 and 4x + 5y = 11, assume we decide to eliminate the y-variable.
Multiply equation 1 by the equation 2 y-coefficient--5--to get: 3(5)x - 7(5)y = 8(5). Simplify to get 15x - 35y = 40. Repeat the process with equation 2. This time, multiply the original equation 2 by -7 to get 4(-7)x + 5(-7)y = 11(-7). Equation 2 simplifies to -28x - 35y = -77. Note that the y-coefficients for both equations are now -35. Multiply one of the equations--it does not matter which one--by "-1." This is to transform one of the "-35" coefficients to a positive 35 for easy cancellation. Using the modified equation 1, the multiplication gives 15(-1)x - 35(-1)y = 40(-1), or -15x + 35y = -40. The equations are ready for elimination.
Add coefficients for matching variables. For x, add -15 and -28. -15 + -28 = -43. For y, 35 + -35 = 0. The numeric constants add to -40 + -77 = -117. The added equation is, therefore, -43x + 0y = -117. Since 0y = 0, the y-variable is eliminated in the addition. The equation summation simplifies to -43x = -117.
Divide by the x-coefficient to find x-value. The result is x = -117/-43 = 2.721. Substitute the x-value into either of the original equations. Using equation 2, 4x + 5y = 11 becomes 4(2.721) + 5y = 11. Subtract the "4(2.721)" term from both sides to get 5y = 11 - 4(2.721), which is 5y = 11 - 10.884, 5y = 0.1163. By same process as finding the x-value, the y-value is y = 0.1163/5 = 0.02325.
Check that solutions fit the other original equation. In this case, equation 1 needs to be checked. 3x - 7y = 8, 3(2.721) - 7(0.02325) = 8.163 - 0.1625 = 8.00025. Very small discrepancies, like that between 8 and 8.00025 are due to round-off error. Since 8.00025 is very close to the ideal value of 8, we can be confident that answers of x = 2.721 and y = 0.02325 are correct.