Linear Equations in Two Variables — Study Notes

1. Introduction

In the earlier classes we studied linear equations in one variable, like 2x + 5 = 9, which have a unique solution. But many real-life situations involve two unknown quantities. For example, if a shopkeeper sells some pens at 5 rupees each and some notebooks at 10 rupees each, and the total bill is 100 rupees, then the number of pens and the number of notebooks are two unknowns connected by one equation. Such situations lead us to linear equations in two variables.

A linear equation in two variables involves two unknown quantities, usually x and y, and each term in the equation contains at most a constant or a single variable raised to the power 1. The general form of such an equation is ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero. In this chapter we will learn how to find solutions of such equations, how to represent them graphically as straight lines, and how to interpret the graph of an equation.

2. General Form and Solutions

The general form of a linear equation in two variables is ax + by + c = 0, where x and y are variables and a, b, c are constants with a and b not both zero.

• If a = 0, the equation becomes by + c = 0, which is a linear equation in one variable y.
• If b = 0, the equation becomes ax + c = 0, a linear equation in one variable x.
• The standard form ax + by + c = 0 is called the general or standard form of the equation.

What is a Solution?

A solution of the equation ax + by + c = 0 is an ordered pair of numbers (x0, y0) which satisfies the equation, meaning that substituting x0 for x and y0 for y makes the left hand side equal to the right hand side. For example, the pair (1, 4) is a solution of 2x + y = 6 because 2(1) + 4 = 6.

Unlike a linear equation in one variable which has a unique solution, a linear equation in two variables has infinitely many solutions. Every ordered pair that satisfies the equation is a solution, and we can generate as many as we like by choosing a value of x and computing the corresponding value of y, or vice versa.

3. Graphical Representation

The most beautiful property of a linear equation in two variables is that its graph is always a straight line. Every point on this line has coordinates which are a solution of the equation, and every solution of the equation is represented by a point on the line.

How to Draw the Graph

1. Write the equation in the form y = (expression in x), if possible.
2. Choose at least two convenient values of x and find the corresponding values of y.
3. Plot the obtained points on a graph sheet.
4. Join the points with a straight line using a ruler, and extend the line beyond the plotted points on both sides.

It is always safer to plot three points so that if one point is wrong due to a calculation error, the error can be detected because the three points will not be collinear.

Parallel and Coincident Lines

• The equation x = a represents a vertical line parallel to the y-axis, passing through x = a on the x-axis.
• The equation y = b represents a horizontal line parallel to the x-axis, passing through y = b on the y-axis.
• The graph of y = mx + c is a straight line; the constant m is called the slope (it measures how steep the line is) and c is the point where the line cuts the y-axis.

4. Equations of the Form x = a and y = b

Every point on the vertical line x = a has the same x-coordinate a, no matter what its y-coordinate is. Similarly, every point on the horizontal line y = b has the same y-coordinate b. These lines are important in geometry because:

• The x-axis is the line y = 0.
• The y-axis is the line x = 0.
• Any line parallel to the y-axis has equation x = a.
• Any line parallel to the x-axis has equation y = b.

For example, the equation x = 4 represents a vertical line passing through the point (4, 0), and y = -2 represents a horizontal line passing through (0, -2). The point of intersection of x = 4 and y = -2 is (4, -2).

5. Applications and Word Problems

Linear equations in two variables help us model real-life situations. When we read a word problem, we must:

1. Identify the two unknown quantities and assign them variables, say x and y.
2. Express the given conditions in the form of an equation.
3. Find a few solutions of the equation.
4. Interpret the solutions in the context of the problem.

For example, "The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables." If the cost of a notebook is x and the cost of a pen is y, then x = 2y, which can be written in the standard form as x - 2y + 0 = 0. If a pen costs 5 rupees, then a notebook costs 10 rupees, and the pair (10, 5) is a solution of the equation.

Quick Revision Tables

Table 1: Special Forms of Linear Equations

Table 2: Number of Solutions

Mind Map

graph TD
    A["Linear Equations in Two Variables"] --> B["General form ax + by + c = 0"]
    A --> C["Solutions"]
    C --> D["Infinitely many ordered pairs"]
    C --> E["Each pair satisfies the equation"]
    A --> F["Graphical representation"]
    F --> G["Graph is always a straight line"]
    F --> H["x = a is vertical, y = b is horizontal"]
    A --> I["Word problems"]
    I --> J["Form the equation from given conditions"]

Important Diagrams (SVG)

Diagram 1: Graph of the Equation 2x + y = 6

Diagram 2: Lines x = a, y = b and the Axes

Common Mistakes

1. Students often say that a linear equation in two variables has only one solution. In fact it has infinitely many solutions, one for every chosen value of x (or y).
2. A common error is writing the solution as (y, x) instead of (x, y). The first number in the ordered pair must be the x-value.
3. Students forget the condition that a and b cannot both be zero in ax + by + c = 0; if both are zero, the equation is not a linear equation in two variables.
4. When drawing the graph, students plot the points and join only two of them, without extending the line. The line must be extended on both sides using a ruler.
5. Students sometimes treat the graph of x = 4 as a horizontal line; it is vertical, parallel to the y-axis.
6. While solving word problems, students assign variables without clearly stating what each variable represents, which leads to wrong equations.
7. Students forget to check whether the obtained pair actually satisfies the original equation before finalising the answer.

Exam Tips

1. Always write the given equation in the standard form ax + by + c = 0 first, because most answer keys expect this form.
2. To find solutions quickly, put x = 0 and y = 0 in turn; these two points are usually the easiest to plot.
3. Plot at least three points and verify they are collinear before drawing the final line, to avoid a crooked graph.
4. For word problems, write down what x and y represent, then form the equation, and finally verify the solution by substituting back.
5. Memorise that x = 0 is the y-axis and y = 0 is the x-axis; these facts are asked repeatedly in exams.
6. In graph questions, label the axes with the variable names and mark the scale clearly, as neatness carries marks.
7. Practise drawing graphs for equations like y = x, y = 2x and x + y = 5 so that the pattern of straight lines becomes intuitive.

Conclusion

In this chapter we learnt what a linear equation in two variables is and its general form ax + by + c = 0. We saw that unlike one-variable equations, a two-variable equation has infinitely many solutions, each an ordered pair (x, y) that satisfies the equation. We studied the graphical representation of such equations and discovered that the graph is always a straight line, with every point on the line giving a solution. We also learnt the special cases: the x-axis is y = 0, the y-axis is x = 0, vertical lines have the form x = a, and horizontal lines have the form y = b. Finally, we learnt how to convert word problems into linear equations and solve them. This chapter connects algebra with coordinate geometry and prepares us for solving pairs of linear equations in Class 10.