Hello students, welcome to today's mathematics lesson. I am so happy to be here with you to learn about a very important chapter in algebra – Chapter 4: Linear Equations in Two Variables. This chapter builds upon what you have already learned about linear equations in one variable, and I promise you, by the end of this lesson, you will have a strong understanding of how equations with two variables work, how to find their solutions, and how to represent them. So let's begin our journey together.
In earlier classes, you studied linear equations in one variable. Do you remember what a linear equation in one variable looks like? Let me give you some examples. Equations like x + 1 = 0, or x + √2 = 0, or even √2 y + √3 = 0 – these are all linear equations in one variable. You learned that such equations have a unique solution, meaning there is only one value that satisfies the equation. For instance, the equation x + 1 = 0 has the solution x = -1, and that's the only possible solution. You also learned how to represent this solution on a number line – a simple point showing where x lies.
Now, in this chapter, we are going to take this knowledge a step further. We are going to extend our understanding from one variable to two variables. This means we will be working with equations that have two unknown quantities instead of just one. Isn't that interesting? You might wonder – why do we need two variables? Well, think about real-life situations. Many problems involve two unknown quantities, and that's where linear equations in two variables become incredibly useful.
Before we dive into the formal definition, let me remind you of some important properties of linear equations that you already know. These properties will be very useful as we work with two variables as well. Are you ready? Here they are:
The solution of a linear equation is not affected when you add the same number to both sides of the equation, or subtract the same number from both sides. Similarly, if you multiply or divide both sides of the equation by the same non-zero number, the solution remains unchanged. These are fundamental properties that hold true for linear equations in one variable, and they will also apply when we work with two variables.
Now, let's consider a real-life situation to understand the need for two variables. Imagine there is a One-day International Cricket match between India and Sri Lanka played in Nagpur. Two Indian batsmen together scored 176 runs. Can we express this information in the form of an equation? Well, we don't know how many runs each batsman scored individually – there are two unknown quantities here. Let us use x to denote the number of runs scored by one batsman, and y to denote the number of runs scored by the other batsman. We know that their total is 176 runs, so we can write:
x + y = 176
This is our equation. And this, students, is an example of a linear equation in two variables. Notice how we used two letters, x and y, to represent the two unknown quantities. It is customary to denote variables in such equations by x and y, but remember that other letters can also be used – like s and t, or p and q, or u and v. The choice of letters is entirely up to us.
Now, let me give you some more examples of linear equations in two variables. Here they are:
1.2s + 3t = 5, p + 4q = 7, πu + 5v = 9, and 3 = √2 x − 7y.
Notice how these equations involve two variables and are linear – meaning the variables are to the power of 1 and are not multiplied with each other.
Now, here's an important point. Each of these equations can be rearranged into a standard form. Can you see how? We can rewrite them as:
1.2s + 3t − 5 = 0, p + 4q − 7 = 0, πu + 5v − 9 = 0, and √2 x − 7y − 3 = 0.
This leads us to the formal definition of a linear equation in two variables. Are you ready for this definition? Listen carefully:
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 is a very important definition, so let me repeat it and explain each part. We have an equation in the form ax + by + c = 0. Here, a, b, and c are real numbers – they can be positive, negative, or zero. The crucial condition is that a and b are not both zero. Why is this condition important? Well, if both a and b were zero, then we would have just c = 0, which is not really an equation in two variables anymore – it would be a constant equation. So we need at least one of a or b to be non-zero to keep it as an equation in two variables.
This means you can think of infinitely many such equations – there is no limit to the number of linear equations in two variables that you can write. As long as you have two variables raised to the power of 1, not multiplied together, and you can rearrange it into the form ax + by + c = 0, it qualifies as a linear equation in two variables.
Now, let me show you some worked examples that will help you understand how to convert equations into this standard form and identify the values of a, b, and c.
Example 1: Write each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, and c in each case:
(i) 2x + 3y = 4.37
(ii) x − 4 = √3 y
(iii) 4 = 5x − 3y
(iv) 2x = y
Let's work through each one together.
For (i): The equation is 2x + 3y = 4.37. To write this in the form ax + by + c = 0, we need to move all terms to one side. So we subtract 4.37 from both sides:
2x + 3y − 4.37 = 0
Now we can identify: a = 2, b = 3, and c = −4.37. Simple, isn't it?
For (ii): The equation is x − 4 = √3 y. Let's rearrange this. First, we bring all terms to one side. We have x on the left, and we want to subtract √3 y and also subtract 4. So we get:
x − √3 y − 4 = 0
Now we can identify: a = 1, b = −√3, and c = −4. Notice how b is negative because we have −√3 y.
For (iii): The equation is 4 = 5x − 3y. This can be written as 5x − 3y − 4 = 0. Here, a = 5, b = −3, and c = −4. But wait, there's an interesting point here. Can this equation also be written differently? What if we multiply the entire equation by −1? Then we would get −5x + 3y + 4 = 0. In this case, a = −5, b = 3, and c = 4. Both representations are correct! This shows us that there can be multiple ways to write the same equation in the form ax + by + c = 0. The key is that the equation itself remains equivalent.
For (iv): The equation is 2x = y. We can rewrite this as 2x − y + 0 = 0. Here, a = 2, b = −1, and c = 0. Notice that c can be zero – that's perfectly fine.
Now, here's something interesting that you should know. Equations of the type ax + b = 0, which you studied in one variable, are also examples of linear equations in two variables. How is that possible? Well, we can express them as ax + 0.y + b = 0. Do you see what we did there? We added a term with y that has a coefficient of zero. For example, the equation 4 − 3x = 0 can be written as −3x + 0.y + 4 = 0. This is now in the form ax + by + c = 0, with a = −3, b = 0, and c = 4. So even though it looks like an equation in one variable, we can always think of it as a linear equation in two variables where the coefficient of one variable is zero. This is a very useful insight.
Let's look at another worked example to practice this concept.
Example 2: Write each of the following as an equation in two variables:
(i) x = −5
(ii) y = 2
(iii) 2x = 3
(iv) 5y = 2
For (i): x = −5 can be written as 1.x + 0.y = −5, or equivalently, 1.x + 0.y + 5 = 0. Here, a = 1, b = 0, and c = 5.
For (ii): y = 2 can be written as 0.x + 1.y = 2, or 0.x + 1.y − 2 = 0. Here, a = 0, b = 1, and c = −2.
For (iii): 2x = 3 can be written as 2x + 0.y − 3 = 0. Here, a = 2, b = 0, and c = −3.
For (iv): 5y = 2 can be written as 0.x + 5y − 2 = 0. Here, a = 0, b = 5, and c = −2.
So you see, any equation that appears to be in one variable can always be expressed as an equation in two variables by adding a term with coefficient zero for the other variable. This is a very flexible way to think about these equations.
Now, let's move on to a very important concept – the solution of a linear equation in two variables. This is where things get really interesting.
You have seen that every linear equation in one variable has a unique solution. But what about linear equations in two variables? Do they also have a unique solution? Let's find out.
As there are two variables in the equation, a solution means a pair of values – one value for x and one value for y – which together satisfy the given equation. We call this an ordered pair, where we first write the value for x and then the value for y. The pair is written in parentheses, like this: (x, y).
Let's consider the equation 2x + 3y = 12. Is x = 3 and y = 2 a solution? Let's check. We substitute x = 3 and y = 2 into the equation:
2x + 3y = (2 × 3) + (3 × 2) = 6 + 6 = 12
Yes, it works! So (3, 2) is a solution of the equation 2x + 3y = 12. We write this as the ordered pair (3, 2).
Now, is (0, 4) also a solution? Let's check: 2(0) + 3(4) = 0 + 12 = 12. Yes, it is! So (0, 4) is another solution.
But what about (1, 4)? Let's check: 2(1) + 3(4) = 2 + 12 = 14, which is not equal to 12. So (1, 4) is NOT a solution. Similarly, note that (0, 4) is a solution but (4, 0) is not – the order matters! In the ordered pair, the first number is x and the second is y. So (4, 0) would mean x = 4 and y = 0, which gives us 2(4) + 3(0) = 8, not 12.
Now, here comes the really interesting part. We have found two solutions: (3, 2) and (0, 4). Can we find any other solutions? What about (6, 0)? Let's check: 2(6) + 3(0) = 12 + 0 = 12. Yes, (6, 0) is also a solution!
But wait, there's more. We can find infinitely many solutions! Let me show you how. Pick any value of x that you like. Suppose we take x = 2. Then the equation 2x + 3y = 12 becomes:
2(2) + 3y = 12 4 + 3y = 12 3y = 12 − 4 = 8 y = 8/3
So (2, 8/3) is another solution. See how we found it? We chose a value for x, then solved for y.
Now let's try another value. Take x = −5. Then:
2(−5) + 3y = 12 −10 + 3y = 12 3y = 22 y = 22/3
So (−5, 22/3) is another solution.
This process can continue forever! There is no end to the different solutions of a linear equation in two variables. This is a fundamental result: A linear equation in two variables has infinitely many solutions. Unlike linear equations in one variable, which have just one solution, equations with two variables have countless solutions. This is a key difference that you must remember.
Now let's look at a worked example to practice finding solutions.
Example 3: Find four different solutions of the equation x + 2y = 6.
We need to find four different ordered pairs (x, y) that satisfy this equation. Let's find them one by one.
First, by inspection – that means by looking at it – we can see that if x = 2 and y = 2, then: x + 2y = 2 + 2(2) = 2 + 4 = 6 So (2, 2) is a solution.
Now, let's find more solutions using a systematic approach. An easy way is to take x = 0 and find the corresponding y. If x = 0, then the equation becomes: 0 + 2y = 6 2y = 6 y = 3 So (0, 3) is a solution.
Similarly, we can take y = 0 and find the corresponding x. If y = 0, then: x + 2(0) = 6 x = 6 So (6, 0) is a solution.
Now let's take y = 1. Then: x + 2(1) = 6 x + 2 = 6 x = 4 So (4, 1) is a solution.
So we have found four solutions: (2, 2), (0, 3), (6, 0), and (4, 1). And remember, these are just four of the infinitely many solutions that exist for this equation. There are countless more!
A remark here: 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. This is a quick method to find two solutions for any linear equation in two variables.
Let's do one more worked example to practice finding solutions.
Example 4: Find two solutions for each of the following equations:
(i) 4x + 3y = 12
(ii) 2x + 5y = 0
(iii) 3y + 4 = 0
For (i): 4x + 3y = 12. Let's take x = 0. Then: 4(0) + 3y = 12 3y = 12 y = 4 So (0, 4) is a solution. Now let's take y = 0. Then: 4x + 3(0) = 12 4x = 12 x = 3 So (3, 0) is another solution.
For (ii): 2x + 5y = 0. Let's take x = 0. Then: 2(0) + 5y = 0 5y = 0 y = 0 So (0, 0) is a solution. Now let's take y = 0 again. We get the same solution (0, 0). To get a different solution, let's take x = 1. Then: 2(1) + 5y = 0 2 + 5y = 0 5y = −2 y = −2/5 So (1, −2/5) is another solution.
For (iii): 3y + 4 = 0. This looks a bit different because there is no x term. But remember, we can write this as 0.x + 3y + 4 = 0. Since there is no x term, x can be anything! Let's see what happens. The equation simplifies to: 3y + 4 = 0 3y = −4 y = −4/3
So y is always −4/3, regardless of the value of x. This means any point with y = −4/3 is a solution. For example, (0, −4/3) is a solution, and (1, −4/3) is also a solution. These are two solutions.
Now, let's summarize everything we have learned in this chapter. This is important because you need to remember these key points for your exams and for future chapters.
In this chapter, you have studied the following points:
First, an equation of the form ax + by + c = 0, where a, b, and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. This is the standard form, and you should always try to write your equations in this form.
Second, a linear equation in two variables has infinitely many solutions. Unlike equations in one variable which have a single unique solution, equations in two variables have an unlimited number of solutions. We can find as many solutions as we want by choosing a value for one variable and solving for the other.
Third, every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the linear equation. This establishes a beautiful connection between algebra and geometry – the solutions of the equation correspond to points on its graph. We will explore this further when we study graphs in subsequent chapters.
These three points are the foundation of this chapter. Make sure you understand them clearly and remember them.
Now, before we conclude, let me quickly recap what we learned today:
We started by recalling linear equations in one variable and then extended the concept to two variables. We learned that a linear equation in two variables can be written in the form ax + by + c = 0, where a and b are not both zero. We practiced converting various equations into this standard form and identifying the values of a, b, and c. We also learned that even equations that look like they are in one variable can be expressed as equations in two variables by adding a term with coefficient zero for the other variable.
Then we moved on to solutions of linear equations in two variables. We learned that a solution is an ordered pair (x, y) that satisfies the equation. We discovered that unlike linear equations in one variable, equations in two variables have infinitely many solutions. We practiced finding multiple solutions by choosing values for one variable and solving for the other. We learned the handy trick of setting x = 0 to find y, and setting y = 0 to find x, which gives us quick solutions.
We also saw some special cases, like when one of the coefficients is zero, which leads to interesting situations where one variable can take any value while the other is fixed.
That's all for today, students. I hope you enjoyed this lesson and found it helpful. Remember, practice is key in mathematics. The more you work with linear equations in two variables, the more comfortable you will become. Thank you for your attention, and I look forward to seeing you in the next lesson. Keep studying hard!