Graphical Representation of Linear Equations in Two Variables

This topic gives an overview of;

- Graph of a Linear Equation in Two Variables
- Equations of Lines Parallel to the x-axis and y-axis

So far, you have obtained the solutions of a linear equation in two variables algebraically. Now, let us look at their geometric representation. You know that each such equation has infinitely many solutions. How can we show them in the coordinate plane? You may have got some indication in which we write the solution as pairs of values. The solutions of the linear equation in Example 3, namely,

This can be expressed in the form of a table as follows by writing the values of y below the corresponding values of x :

In the previous chapter, you studied how to plot the points on a graph paper. Let us plot the points (0, 3), (2, 2), (4, 1) and (6, 0) on a graph paper. Now join any two of these points and obtain a line. Let us call this as line AB.

Do you see that the other two points also lie on the line AB? Now, pick another point on this line, say (8, –1). Is this a solution? In fact, 8 + 2(–1) = 6. So, (8, –1) is a solution. Pick any other point on this line AB and verify whether its coordinates satisfy the equation or not. Now, take any point not lying on the line AB, say (2, 0). Do its coordinates satisfy the equation? Check, and see that they do not.

Let us list our observations:

- Every point whose coordinates satisfy Equation (1) lies on the line AB.
- Every point (a, b) on the line AB gives a solution x = a, y = b of Equation (1).
- Any point, which does not lie on the line AB, is not a solution of Equation (1).

So, you can conclude that every point on the line satisfies the equation of the line and every solution of the equation is a point on the line. In fact, a linear equation in two variables is represented geometrically by a line whose points make up the collection of solutions of the equation. This is called the graph of the linear equation. So, to obtain the graph of a linear equation in two variables, it is enough to plot two points corresponding to two solutions and join them by a line. However, it is advisable to plot more than two such points so that you can immediately check the correctness of the graph.

The reason that a, degree one, polynomial equation ax + by + c = 0 is called a *linear equation* is that its geometrical representation is a straight line.

You have studied how to write the coordinates of a given point in the Cartesian plane. Do you know where the points (2, 0), (–3, 0), (4, 0) and (n, 0), for any real number n, lie in the Cartesian plane? Yes, they all lie on the x-axis. But do you know why? Because on the x-axis, the y-coordinate of each point is 0. In fact, every point on the x-axis is of the form (x, 0). Can you now guess the equation of the x-axis? It is given by y = 0. Note that y = 0 can be expressed as 0. x + 1.y = 0. Similarly, observe that the equation of the y-axis is given by x = 0.

Now, consider the equation x – 2 = 0. If this is treated as an equation in one variable x only, then it has the unique solution x = 2, which is a point on the number line. However, when treated as an equation in two variables, it can be expressed as x + 0.y – 2 = 0. This has infinitely many solutions. In fact, they are all of the form (2, r), where r is any real number. Also, you can check that every point of the form (2, r) is a solution of this equation. So as, an equation in two variables, x – 2 = 0 is represented by the line AB in the graph.

- The graph of every linear equation in two variables is a straight line.
- x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
- The graph of x = a is a straight line parallel to the y-axis.
- The graph of y = a is a straight line parallel to the x-axis.
- An equation of the type y = mx represents a line passing through the origin.
- 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.

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