![]() ![]() Unique solution (consistent and independent) a1/a2 ≠ b1/b2.You can check directly about the types of solutions using the following conditions: ![]() In fact, it is the only solution to the pair, as two non-parallel lines cannot intersect at more than one point. The point of intersection is (1,3), which means that x = 1, y = 3 is a solution to the pair of linear equations given by (2). We draw the corresponding lines on the same axes: ![]() Thus, the non trivial solution is: x = 1, y = 3Īs an example, lets solve the following linear equation: x - y + 2 = 0, 2x + y - 5 = 0. Once we have the value of y, we proceed as earlier – we plug this into any of the two equations. Note how x gets eliminated, and we are left with an equation in y alone. Now, let us subtract the two equations, which means that we subtract the left-hand sides of the two equations, and the right-hand side of the two equations and the equality will still be preserved.Ħx + 9y - 33 = 0 ,6x + 4y - 18 = 0 0 + 5y - 15 = 0, 5y = 15, y = 3 ![]() Let us multiply the first equation by 3 and the second equation by 2, so that the coefficients of x in the two equations become equal: The coefficients of x in the two equations are 2 and 3 respectively. We express one variable in terms of another using one of the pair of equations and substitute that expression into the second equation.Ĭonsider the following pair of linear equations: It should be clear why this process is called substitution. The final non trivial solution is: x = 3, y = 1.Lets plug it into the first equation: x + y = 4 (3) + y = 4, y = 4 - 3 = 1, y = 1 Once we have the value of x, we can plug this back into any of the two equations to find out y.This expression for y can now be substituted in the second equation, so that we will be left with an equation in x alone: x - y = 2, x - 4 + x = 2, 2x = 6 x = 6/2, x = 3.Let’s rearrange the first equation to express y in terms of x, as follows: x + y = 4, y = 4 - x.The following methods can be used to find the solutions of linear equations of two variables.Ĭonsider the following pair of linear equations, let's solve the following linear equations. Solutions for Linear Equations in Two Variables Hence, the solution of the equation 2x + 4 = 8 is x=2. Now we have to remove 2 from L.H.S in order to get x, therefore we divide the equation by 2.To find the value of x, first, we remove 4 from L.H.S, so we subtract 4 from both sides of the equation.How to find the Solution of a Linear Equation? Solutions for Linear Equations in One Variable ![]()
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