Solving For X Problems

When solving a simple equation, think of the equation as a balance, with the equals sign (=) being the fulcrum or center.

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This is another very easy and useful equation solving technique that is extensively used in Algebraic calculations. In this example, we see that neither the coefficients of x nor those of y are equal in the two equations.

So simple addition and subtraction will not lead to a simplified equation in only one variable.

And that value is put into the second equation to solve for the two unknown values.

The solution below will make the idea of Substitution clear. x y = 15 -----(2) (10 y) y = 15 10 2y = 15 2y = 15 – 10 = 5 y = 5/2 Putting this value of y into any of the two equations will give us the value of x.

x y = 24 ------(2) (we find the value of x in terms of y) x = 24 –y (Next we put this value of x into equation (1)) 2(24 – y) – 2y = –2 48 – 2y – 2y = –2 48 – 4y = – 2 (Subtracting 48 from both sides of the equation gives) 48 – 4y – 48 = –2 –48 –4y = –50 (Dividing on both sides of the equation by – 4) -4y/-4 = -50/-4 y = 50/4 = 25/2 (Putting this value of y into equation (2) and then solving for x gives) x 25/2 = 24 (Subtracting 25/2 from both sides of the equation gives) x 25/2 - 25/2 = 24 - 25/2 x = (48 - 25)/2 = 23/2 Hence the solution to the given system of equations is (x , y) = ( 23/2 , 25/( 2 )) Note: Next we show what happens if we substitute the value of x into the same equation that we used to compute it (equation (2) in this example) x y = 24 24 – y y = 24 ∵ (x = 24 – y) 24 = 24 This is the result that we are left with.

There is nothing wrong with 24 being equal to 24, but then what should we do with it?x y = 15 x 5/2 = 15 x = 15 – 5/2 x = 25/2 Hence (x , y) = (25/2, 5/2) is the solution to the given system of equations. In Elimination Method, our aim is to "eliminate" one variable by making the coefficients of that variable equal and then adding/subtracting the two equations, depending on the case.In this example, we see that the coefficients of all the variable are same, i.e., 1.0≠ –2 Hence the two equations constitute an inconsistent system of linear equations and thus do no have a solution (At no point do the two straight lines intersect = In this method of equation solving, we work out on any of the given equations for one variable value, and then substitute that value in the other equation.It gives us an equation in a single variable and we can use a single variable equation solving technique to find the value of that variable (as shown in examples above).So if we add the two equations, the –y and the y will cancel each other giving as an equation in only x. x – y = 10 x y = 15 2x = 25 x = 25/2 Putting the value of x into any of the two equations will give y = 5/2 Hence (x , y) = (25/2, 5/2) is the solution to the given system of equations.Elimination Method - By Equating Coefficients: In Elimination Method, our aim is to "eliminate" one variable by making the coefficients of that variable equal and then adding/subtracting the two equations, depending on the case.Doing the Sometimes you have to use more than one step to solve the equation.In most cases, do the addition or subtraction step first.There will be no change in the equation solving strategy and once you have learnt the above method, you do not need to bother about the coefficients at all.Next we present and try to solve the examples in a more detailed step-by-step approach.

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