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How to Solve Linear Equtions By Subtraction

Solving a set of linear equations means finding a set of values for the variables that fit all of the equations. If you graph the lines that represent the equations, the solution will be the point where all the lines meet. Often graphing is too much trouble -- and graphs can be imprecise -- so other simpler methods are used to find the point where the lines cross. This can sometimes be done by subtracting the equations.

Instructions

    • 1

      Recast the formulas in standard form -- so they look like aX + bY = c where a, b and c are numbers. Often linear equations are in the slope-intercept form Y = mX + b, which is not good for solving by substitution. The thing that makes the linear equations difficult to solve is that there are two or more variables in each equation. We want to eliminate all but one of the variables so that we have an equation with one variable, which will be easy to solve. Once we have a value for one variable, we can go back to either of the original equations and substitute the newly found value to get the value of the other variables. If there are more than two variables you may have to do this several times.

    • 2

      Multiply each equation by numbers that make the coefficient of one of the terms -- either the X terms or the Y terms -- equal. For example, if the equations are 5X + 3Y = 11 and 7X - 4Y = -1, multiply the first equation by 4 and the second by -3 to get 20X + 12Y = 44 and -21X +12Y = 3. Now the Y coefficient is the same in both equations.

    • 3

      Subtract -21X + 12Y = 3 from 20X + 12 = 44 to get 41X = 41. This means that X = 1. Going back to the 5X + 3Y = 11 equation and substituting in the newly found value for X, we have 5(1) + 3Y = 11 or 3Y = 6, which implies Y = 2. This means that the equations 5X + 3Y = 11 and 7X - 4Y = -1 intersect at the point (1,2).

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