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Linear Equations in Two Variables

    Solution of a Linear Equation

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9th class > Linear Equations in Two Variables > Solution of a Linear Equation

Solution of a Linear Equation

You have seen that every linear equation in one variable has a unique solution. What can you say about the solution of a linear equation involving two variables? As there are two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation.

Let us consider the equation 2x + 3y = 12. Here, x = and y = is a solution.

Because when you substitute x = 3 and y = 2, in the equation above, you find that:

2x + 3y = (2 × 3) + (3 × 2) = + =

This solution is written as an ordered pair (3, 2), first writing the value for x and then the value for y. Similarly, (0, 4) is also a solution for the equation above.

On the other hand, (1, 4) a solution of 2x+3y=12.

This is because on putting x = 1 and y = 4 we get 2x + 3y = 14, which is not 12. Note that (0, 4) is a solution but not (4, 0).

You have seen at least two solutions for 2x + 3y = 12, i.e., (3, 2) and (0, 4). Can you find any other solution? Do you agree that (6, 0) is another solution?

In fact, we can get many solutions in the following way. Pick a value of your choice for x (say x = 2) in 2x+3y=12. Then the equation reduces to 4+3y=12 which is a linear equation in one variable.

On solving this, you get y = .

So, is another solution of 2x + 3y = 12. Similarly, choosing x = – 5, you find that the equation becomes –10 + 3y = 12. This gives y = .

So, −5,223 is another solution of 2x + 3y = 12. So there is no end to different solutions of a linear equation in two variables. That is, a linear equation in two variables has infinitely many solutions.

Example 3

Find four different solutions of the equation x + 2y = 6.

Solution:

By inspection, x = 2, y = 2 is a solution because:

x + 2y = 2 + 4 =

Now, let us choose x = 0. With this value of x, the given equation reduces to 2y = 6 which has the unique solution of y = .

So 0,3 is also a solution of x+2y=6.

Similarly, taking y = 0, the given equation reduces to x = .

So, (6,0) is a solution of x+2y=6 as well.

Finally, let us take y = 1. The given equation now reduces to x + 2 = 6, whose solution is given by x = .

Therefore, (4, 1) is also a solution of the given equation.

So four of the infinitely many solutions of the given equation are: (2, 2), (0, 3), (6, 0) and (4, 1).

Remark :

Note that an easy way of getting a solution is to take x = 0 and get the corresponding value of y. Similarly, we can put y = 0 and obtain the corresponding value of x.

Example 4

Find two solutions for each of the following equations:

(i)

(i)4x + 3y = 12

Solution

Taking x = 0, we get 3y = 12, i.e., y = .

So, (0, 4) is a solution of the given equation.

Similarly, by taking y = 0, we get x = .

Thus, (3, 0) is also a solution.

(ii)

(ii)2x + 5y = 0

Solution

Taking x = 0, we get 5y = 0, i.e., y = .

So (0, 0) is a solution of the given equation.

Now, if you take y = 0, you again get (0, 0) as a solution, which is the same as the earlier one.

To get another solution, take x = 1, say. Then you can check that the corresponding value of y is .

So 1,−25 is another solution of 2x+5y=0.

(iii)

(iii)3y + 4 = 0

Solution

Writing the equation 3y+4=0 as x + y + = 0, you will find that y = for any value of x.

Thus, two solutions can be given as 0,−43 and 1,−43.

Find the price of variables that satisfy the following equation:

2x + 5y = 20 and 3x + 6y = 12.

Instructions

Find the price of variables

  • Using the method of substitution to solve the pair of linear equation, we have first eqn: 2x + 5y = 20 (i)
  • The second eqn: 3x + 6y = 12 (ii)
  • Multiplying equation (i) by 3 and then we get: x + y = (iii)
  • Multiplying equation (ii) by 2, and then we get : x + y = (iv)
  • Subtracting equation (iv) from (iii) we get: y =
  • Therefore, we get: y =
  • Substituting the cost of y in any of the equations (i) or (ii), we've apply the y value in equation (i)
  • Calculating the values
  • Therefore, we get: x =
  • Consequently, x = and y = is the factor where in the given equations intersect.