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In earlier classes, you have studied linear equations in one variable. Write down a linear equation in one variable. You may say that x + 1 = 0, x + √2 = 0 and √2 y + √3 = 0 are examples of linear equations in one variable. You also know that such equations have a unique (i.e., one and only one) solution. You may also remember how to represent the solution on a number line. In this chapter, the knowledge of linear equations in one variable shall be recalled and extended to that of two variables. You will be considering questions like: Does a linear equation in two variables have a solution? If yes, is it unique? What does the solution look like on the Cartesian plane?
Let us first recall what you have studied so far. Consider the equation:2x + 5 = 0. Its solution, i.e., the root of the equation, is -5/2. This can be represented on the number line as shown below:
This can be represented on the number line as shown below:
While solving an equation, you must always keep the following points in mind: The solution of a linear equation is not affected when:
Let us now consider the following situation: In a One-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Express this information in the form of an equation.
Here, you can see that the score of neither of them is known, i.e., there are two unknown quantities. Let us use x and y to denote them. So, the number of runs scored by one of the batsmen is x, and the number of runs scored by the other is y. We know that x + y = 176 which is the required equation.
This is an example of a linear equation in two variables. It is customary to denote the variables in such equations by x and y, but other letters may also be used. Some examples of linear equations in two variables are:
1.2s + 3t = 5, p + 4q = 7, πu + 5v = 9 and 3 = 2 x – 7y
Note that you can put these equations in the form 1.2s + 3t – 5 = 0, p + 4q – 7 = 0, πu + 5v – 9 = 0 and 2 x – 7y – 3 = 0, respectively.
So, any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables. This means that you can think of many many such equations.
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 = 3 and y = 2 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) = 12
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) is not a solution of 2x + 3y = 12, 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? Verify the same. In fact, we can get many 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 . so is another solution of .
Similarly, choosing x = – 5, you find that the equation becomes –10 + 3y = 12.
This gives so is another solution of .
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.
Let us consider an example to find four different solutions of the equation x + 2y = 6.
By inspection, x = 2, y = 2 is a solution because for x = 2, y = 2 x + 2y = 2 + 4 = 6 which has the unique solution y = 3. So x = 0, y = 3 is also a solution of x + 2y = 6. Similarly, taking y = 0, the given equation reduces to x = 6. So, x = 6, y = 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 = 4. Therefore, (4, 1) is also a solution of thegiven equation. So four of the infinitely many solutions of the given equation are: (2, 2), (0, 3), (6, 0) and (4, 1).
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.
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