Read till taught in 8th. ## Linear Equation in two variablesLet us first look at the form of linear equations in two variables.ax + by = c Such equations arise when we have two things changing at the same time. The simplest is to watch it on a graph. The x-axis forms one variable and the y-axis forms the other variable. A line has infinite number of solutions but we have to get a unique solution. And this is possible only when the number of lines is two and they intersect or in other words we have two linear equations in two variable. Hence to solve a system of two variable linear equation the required equations are dx + ey = f Substitution method or by Elimination method.Substitution methodIn substitution method we find the value of one variable in terms of the other and substitute it in the other equation. ax + by = c ----(i)dx + ey = f ----(ii) From (i) x = (c - by)/aSubstituting in second we get d(c - by)/a + ey = f dc/a - dby/a + ey= f dc/a - (db - ea)y/a = f dc - (db - ea)y = fa (db - ea)y = (dc - fa) y = (dc - fa)/(db - ea) Substituting in (i) we get ax + b(dc - fa)/(db - ea) = c ax = c - b(dc - fa)/(db - ea) ax = [c(db - ea) - b(dc - fa)]/(db - ea) x = [cdb - cea - bdc + bfa]/a(db - ea) x = (bf - ce)a/a(db - ea) x = (bf - ce)/(db - ea) x = (bf - ce)/(db - ea) = (ce - bf)/(ea - db)y = (dc - fa)/(db - ea) = (fa - dc)/(ea - db) Elimination methodIn elimination method we eliminate one variable by equaling the other variable in both the equations. Then we substitute the value of first variable to get the value of other variable. ax + by = c ----(i)dx + ey = f ----(ii) Multiplying (i) by e and (ii) by b, we getaex + bey = ce ----(iii) dbx + bey = fb ----(iv) Subtracting (iv) from (iii) we get (ae - db)x = (ce - fb) x = (ce - fb)/(ae - db) As substituted above in substitution method we get, y = (fa - dc)/(ea - db) x = (ce - bf)/(ea - db)y = (fa - dc)/(ea - db) |

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