In the previous article Types of Linear Equations you might have understood the basics of Linear Equations and their types. In this section you will find the methods to solve linear equations.
rnrn
There are two methods to solve linear equations:
rnrn
In the first method, we eliminate (remove) the other variables from the given set of equations to find the values of one variable. The step-wise process for finding the value of variables by elimination method is given below.
rnStep 1: Choose the variable(s) to be eliminated.
rnStep 2: Multiply the equations with values such that the coefficients of the variables (that are to be eliminated) in both the equations are same.
rnStep 3: Add or subtract the equations to eliminate the variables and find the values of the other variables.
rnStep 4: Substitute the values of the variables found in the previous step in all the equations and solve for the other variables.
rnrn
Let us see an example and find the variable using the above steps,
rnIf x + 2y = 8 and 2x – 5y = 19, find the value of y?
rnThe equations are,
rnx + 2y = 8 …… (1)
rn2x + 5y = 19 …… (2)
rnrn
rn
From the given equations, we need to find the value of y. Hence x has to be eliminated. To eliminate x, the coefficients of x in both the equations have to be the same. For this to happen, equation (1) has to be multiplied by 2 and equation (2) has to be the same.
rn2x + 5y = 19 …… (2)
rnEquation (1): multiply with 2 → 2x + 4y = 16 …… (3)
rnSubtracting (3) from (2), we get y = 3.
rnIf the value of x also has to be determined, substitute the value of y in (1) and solve for it.
rnrn
Another method of solving linear equations is by substitution from answer options, and this is the trial and error method.
rnIn this method, we try to find the value of a variable by substituting the answer options in the equations framed from the given information and checking for consistency. This iteration is repeated with all the answer options till one of them is consistent with all the given information.
rnrn
Let us take this example If 3x + 2y = 16 and 5x + 7y = 45, find the value of x?
rnAnswer options are:
rna. 5
rnb. 2
rnc. 3
rnd. 4
rnrn
From the first equation, x has to be an even number to satisfy the equation (2y will be an even number and only the sum of two even numbers can be 16). From the given answer options, the answer can be 2 or 4.
rnSubstituting x = 2 in the first equation, the value of y = 5.
rnSubstituting x = 2, y = 5 in the second equation, the equation holds true. Hence the value of x is 2.
rnrn
The trickiest thing about these methods are they look simple. However, don’t let this stop you from practicing them well. All the best!