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Are you preparing coordinate geometry? If yes, then the most important concept of geometry is graphing linear equations. This topic plays a vital role while preparing for the exams. So, check out the useful information about the linear equations, graphing linear equations in two variables, and solved example questions from the following sections.

## Linear Equations Definition

A linear equation is a first-order equation which is having the power of the variable as one. If an equation has only one variable, then it is called the linear equations in one variable. When the linear equations have two variables, it is known as the linear equations in two variables, etc. The standard form the linear equations in two variables is ax + by + c = 0.

Here,

a, b, c are the real numbers

a, b is not equal to zero.

The solution of a linear equation in two variables is a pair of numbers, they are x, and y which satisfies the equation. The example equation is x + 3y = 12. Some of the solutions are (0, 4), (12, 0), (3, 3), (2, 6).

Properties for the graphing linear equation:

- Every linear equation has infinite solutions.
- Every point (x, y) on the line gives the solutions.
- Every point on the line satisfies the equation.
- To draw an exact line on the graph paper you can plot as many points you like, but it is necessary to plot a minimum of three points.

Therefore, every linear equation in two variables can be represented geometrically as a straight line in a coordinate plane.

### How to Graph of Linear Equation in Two Variables?

Below provided are the simple steps that are helpful to you to draw a linear equation in a graph. Go through those steps and use them.

- Convert the equation in the form of y = mx + c (slope-intercept form)
- Use the trial and error method to get 3 pairs of points that satisfy the equation.
- Plot those three points on a graph.
- Join those points marked on the graph paper to get the straight line that represents the equation graphically.

### Example Questions

1. Draw the graph of the linear equation y = 5x?

Solution:

Given linear equation is y = 5x

Equation is already available in the form of y = mx + c [here c = 0]

Now we will apply the trial and error method to find 3 pairs of values of (x, y) which satisfy the given equation y = 5x.

When the value of x = 0, then y = 5 × 0 = 0

When the value of x = 1, then y = 5 × 1 = 5

When the value of x = -1, then y = 5 × -1 = -5

Arrange the values of the linear equation y = 5x in the table.

Linear Equation Table is

x | 0 | 1 | -1 |
---|---|---|---|

y | 0 | 5 | -5 |

Now, plot the points P (0, 0), Q (1, 5), R (-1, -5) on the coordinate graph

Join those points to get a straight line.

2. Draw the graph of the linear equation y = x + 2.

Solution:

Given linear equation is y = x + 2

Equation is already available in the form of y = mx + c [here c = 2, m = 1]

Now we will apply the trial and error method to find the values of ordered pairs that satisfy the given equation y = x + 2.

When the value of x = -2, then y = -2 + 2 = 0

When the value of x = -1, then y = -1 + 2 = 1

When the value of x = 0, then y = 0 + 2 = 2

When the value of x = 1, then y = 1 + 2 = 3

When the value of x = 2, then y = 2 + 2 = 4

Arrange the values of the linear equation y = x + 2 in the table.

Linear Equation Table is

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

y | 0 | 1 | 2 | 3 | 4 |

Now, plot the points A (-2, 0), B (-1, 1), C (0, 2), D (1, 3), E (2, 4) on the coordinate graph

Join those points to get a straight line.

3. Draw the graph of the linear equation y = 3x − 6.

Solution:

Given linear equation is y = 3x – 6

It is in the slope-intercept form y = mx + c [ where m = 3, c = -6]

Now, apply the trial and error method to get the values of the points that satisfy the equation y = 3x – 6

When the value of x = -1, then y = 3(-1) -6 = -3 – 6 = -9

When the value of x = 0, then y = 3(0) – 6 = -6

When the value of x = 1, then y = 3(1) – 6 = 3 – 6 = -3

When the value of x = 2, then y = 3(2) – 6 = 6 – 6 = 0

Arrange the values of the linear equation y = 3x – 6 in the table.

x | -1 | 0 | 1 | 2 |
---|---|---|---|---|

y | -9 | -6 | -3 | 0 |

Now, plot the points A (-1, -9), B (0, -6), C (1, -3), D (2, 0) on the coordinate graph

Join those points to get a straight line.