Linear equations in two variables

Linear equations in two variables: Welcome you all to our newest blog on the "linear equations in two variables". In today's blog, you will get the complete knowledge of linear equations. This is an very important topic for every exams like Civil services exam.

 So now without wasting much time let's start. 

• Linear equations in two variables

Linear Equations in Two Variables

Find two numbers whose sum is 14. Using variables 'x' and 'y' for the two numbers, we can form the equation x + y = 14. This is an equation in two variables. We can find many values of x and y satisfying the condition. 

E.g.  9+5 = 14

Conventionally, the solution x=9 and y=5 is written as (9,5), where 9 is the value of x and 5 is the value of y. To satisfy this equation x+y=14, we can get infinite ordered pairs like (9,5), (7,7), (S,6), (4,IO), (l0,4), (-1,15), (2.6, 11.4), ... etc. All of these are the solutions of x + y = 14. 

Consider the second example. Find two numbers such that their difference is 2. Let the greater number be x and the smaller number be y. Then we get the equation x -y = 2 for the values of x and y, we can get following equations. 

E.g. 10 - 8 =2

         9 - 7 = 2

         8 - 6 = 2

        (-3) - (-5) = 2 

         5.3 - 3.3 = 2 

         

Here if we take values x = IO and y = 8, then the ordered pair ( 1 O, 8) satisfies the above equation. Here we cannot write as (8, 10) because (8, 1 O) will imply x = 8 and y = IO and it does not satisfy the equation x -:-y = 2. Therefore, note that the order of numbers in the pair indicating solution is very important. 

Math 1 - Algebra

• Remember this!

When we consider two linear equations in two variables simultaneously and we get the unique common solution, then such set of equations is known as Simultaneous equations.  

• Eliminating method of solving simultaneous equations

By taking different values of variables we have solved the equations x + y = 14 and x -y = 2. But every time, it is not easy to solve by this method, e.g. , 2.x + 3y = -4 and x -Sy= 11.        Try to solve these equations by taking different values of x .and y. By this method observe that it is not easy to obtain the solution. Therefore to solve simultaneous equations we use different method. In this method, we eliminate one of the variables to obtain equations in one variable. We can solve and find the value of one of the two variables and then substituting this value in one of the given equations we can find the value of the other variable. Study the following example to understand this method.

(Ex.) Some of the ages of mother and son is 45 years. If son's age is subtracted from twice of mother's age then we get to answer 54. Find the ages of mother and son. 

Answer = It becomes easy to solve a problem if we make use of variables. 

Solution : 

Let the mother's today's age be x years and son's today's age by y years.

From the first condition x+y...........I 

From the second condition 2x-y = 54 .........II

Adding equations (I) and (II) 

∴ 3x+0 = 99

∴     3x = 99 

∴      x  = 33

Substituting x = 33 in equation (I), 

∴ 33+y = 45 

∴        y = 45 - 33 

∴        y = 12 

Verify that x = 33 and y = 12 is the solution of the second equation. Today's age of the mother is 33 and that the son is 12 years. 

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