Graphing linear equations is pretty simple, however requires neat work to be done.
The standard form of a linear equation is y = mx + b known as the Slope Intercept form.
where m is the slope and b is the y-intercept.

The coordinate system consists of a graph containing an x-axis and y-axis. Each axis will have positive and negative numbers. A point on a graph has an x-value and a y-value written as (x, y) known as coordinates of a point.Coordinates are just ordered pairs of numbers for example, (1, 7), ( -3, 8) etc.,
Coordinates always follow this order:
(x, y) <-- It's alphabetical!

To plot the point with the coordinate (7, 6) follow along the x-axis until you get 7 and then go up until you get across from the 6 on the y -axis.This is how we place the point.
Graph Plotting Points
When there is only one variable in the equation the line can either be horizontal or vertical.
Consider an equation: x = - 8

The given equation is independent of y in the xy - plane.

Given below is the plot for x = -8.
Linear Graph
A linear equation in three variables is of the form : ax + by + cz = d, where x, y and z should not be zero.
When you graph a linear equation in 3 variables, instead of a line, you get a triangular plane – three connected lines and everything inside them.

To graph a linear equation in three variables, you find the three intercepts, graph them, then connect them.

For example, Consider the equation 3x + 2y + 4z = 12
The points are x = 4, y = 6 and z = 3
Then the coordinates are :(4, 0, 0), (0, 6, 0), (0, 0, 3)
The graph of linear equation with three variables will represent a plane in 3D space.
Slope- Intercept form is most commonly used to represent linear equations and is of the form y = mx + b where m is the slope and b is the y intercept.

Let's plot y = $\frac{2}{3}$x - 4

We see that for the given equation m = $\frac{2}{3}$ and b = -4

We get y values when x values are substituted randomly in the equation.

x
y
-1 -4.67
-2 -5.33
0 -4
1 -3.33
2 -2.66

The graph is given below:
Linear Equation Graph
While graphing linear equations involving fractions, it is good to clear fractions first by multiplying by the least common denominator of all the fractions involved. Simplify them by taking remaining to the other side.
Consider: $\frac{2}{3}$x + 6 = 1

Given: $\frac{2}{3}$x + 6 = 1

3($\frac{2}{3}$x + 6) = 3(1)

2x + 18 = 3

2x = -15

$\Rightarrow$ x = -7.5
The symbols <, >, $\leq$, $\geq$ denote inequalities.
Graph the region for the linear inequality y $\leq$ -2x + 3

The given equation represents Slope-Intercept form with m = -2 and c = 3

We can solve for the given points algebraically using the slope formula.

$\frac{y_{2} - y_{1}}{x_{2}-x_{1}}$ = m

Plug in $x_{2}$ = 0 and $y_{2}$ = 3 ($y_{1}$ = y, $x_{1}$ = x)

$\frac{3 - y}{0 - x}$ = $\frac{-2}{1}$

3 - y = -2
$\Rightarrow$ y = 5

and -x = 1
$\Rightarrow$ x = -1

So now we have the point (-1, 5)

To find another point plug in $y_{1}$ = -3 and $x_{1}$ = 0
($y_{2}$ = y, $x_{2}$ = x) we get

$\frac{y -3}{x - 0}$ = $\frac{-2}{1}$

y - 3 = -2 $\Rightarrow$ y = 1

x - 0 = 1 $\Rightarrow$ x = 1

The other point is (1, 1) .

Given below is the graph for the given equation:

Linear Equation
From the graph we see the points are same so the answer is verified. The shaded region gives the solution to the given problem.

Solved Examples

Question 1: Graph the region for the linear inequality  y $\leq$ 5x + 2
Solution:
 
The given represents an slope intercept form where m = 5 and c = 2
The equation is taken as y = 5x + 2
The Graph is given below.
Slope Intercept Form
The shaded region is the solution for the given problem.

 

Question 2: Solve: y =  9x - 4
Solution:
 
It is clear that the given equation is in slope intercept form  y = mx + c where m = 9 and c = -4
Putting x = 0, 1, 2 in the given equation we get y values and are tabulated below.

    x        y
    0   -4
    1     5
    2    14

Intercept Form