LINEAR GRAPHS

Example :

An equation such as x - 2y = -2 is an example of a linear equation in the two variables x and y.

A solution of this equation is an ordered pair of numbers (x₀, y₀) so that x₀ and y₀  satisfy the equation x - 2y = -2, in the sense that x₀ - 2y₀ = -2.

Using the above method of getting all the solutions of the equation x - 2y = -2, we can plot as many points of the graph as we please to get a good idea of the graph.

For example, the adjacent picture contains the following points (given by the dots) on the graph, going from left to right :

(-5, -1.5), (-4, -1), (-2, 0), (0, 1), (2, 2), (4, 3), (6, 4).

These points strongly suggest that the graph of x - 2y = -2 is a straight line.

General Equation of Linear Graphs

Thus, a first degree equation in two variables always represents a straight line. Hence we can take general equation of a straight line as ax + by + c = 0, with at least one of a or b not equal to zero.

For the sake of simplicity to draw lines in graphs we consider y = mx + b. as another simple form of the equation of straight line. For each value of x, the equation y = mx + b gives a value of y and we can obtain an ordered pair (x, y).

Important Note

(i) If c = 0, then the equation becomes ax + b = 0 and the line passes through the origin.

(ii) If a = 0, then the equation becomes by + c = 0 and the line is parallel to x-axis.

(iii) If b = 0, then the equation becomes ax + c = 0 and the line is parallel to y-axis.

Procedure to Draw a Linear Graph

When graphing an equation, we usually begin by creating a table of x and y values. We do this by choosing three x values and computing the corresponding y values.

Although two points are sufficient to sketch the graph of a line, we usually choose three points so that we can check our work.

Step 1 :

Using the given equation construct a table of with x and y values.

Step 2 :

Draw x-axis and y-axis on the graph paper.

Step 3 :

Select a suitable scale on the coordinate axes.

Step 4 :

Plot the points

Step 5 :

Join the points and extend it to get the line.

Drawing Linear Graph

Draw the graph of the line joining the points

(3, 5) and (-5, 1)

Solution :

Step 1 :

Draw the x-axis and y-axis on a graph sheet with 1 cm = 1 unit on both axes.

Step 2 :

We plot the two given points (3, 5), (-5, 1) on the graph sheet.

Step 3 :

We join the points by a line segment and extend it on either direction.

Step 4 :

We get the required linear graph.

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