# Linear Equations

So far, all the equations that we have come across are linear in type. The most common difference between the two types of equations is as follows:

### Linear Equations

•  A simple linear equation is of the form: y = mx + c
•  A linear equation looks like a straight line when graphed.
•  It has a constant slope value.
•  The degree of a linear equation is always 1.
•  Superposition principle is applicable to a system characterized by a linear equation.
•  The output of a linear system is directly proportional to its input.

### Non-Linear Equations

•  A simple non-linear equation is of the form: ax2 + by2 = c
•  A non-linear equation look like a curve when graphed.
•  It has a variable slope value.
•  The degree of a non-linear equation is at least 2 or other higher integer values. With the increase in the degree of the equation, the curvature of the graph increases.
•  Superposition principle does not apply to the systems characterized by non-linear equations.
•  The input and output of a non-linear system is not directly related.

We have learned some techniques to solve linear equations. Solutions to non-linear equations are also possible, but they are comparatively difficult and more involved.

## Some interesting Topics related to Equations:

Next we discuss a few interesting things about equations. Kids your age might wonder as to how they can draw:

•  A simple linear equation is of the form: y = mx + c
•  A linear equation looks like a straight line when graphed.

on a piece of paper. We learn some easy ways to graph a linear equation in one or two variables.

## Graphing a linear equation in one variable:

Graphing an equation requires a co-ordinate plane. It consists of two straight lines one in horizontal direction and the other in the vertical direction. The horizontal line is referred to as x-axis and the vertical line is called y-axis. The point where the two lines intersect is called origin.

A simple coordinate plane has been shown below. There exist infinitely many points on the coordinate plane. A single point can be specified with the help of two co-ordinate values x and y, and is represented in the form of an ordered pair (x,y). Here x and y can take any real value.

In order to graph a linear equation in one variable, we make use of a coordinate plane Let us present it through an example.

## Plot the linear equation x – 3 = 0 and y + 2 = 0 on the graph.

#### Explanation:

The given equation is

x – 3 = 0
x = 3
y+ 2 = 0
y = –2

This can be plotted on the coordinate plance as shown below. Next we prsenet the graph of an equation in two variables.

Graphing a linear equation in two variables:

This is easy and much similar to the above method of graphing. Let us present it through an example.

## Plot an equation y = 2x + 3 on the coordinate plane.

#### Explanation:

The given equation is y = 2x + 3. Since the equation has two variables x and y, we take two random values of x, and calculate the corresponding values of y by putting x into the equation.

Let us take x = 1 and x = –1.

x              y

+1       2(+1) + 3 = 5
–1      2(–1) + 3 = 1

Now we plot the two points (1,5) and (–1,1) on the graph as shown in the figure below. Now you can simply join these two points by a straight line and that will give you the required graph of the given equation.

You can also varify that the graph obtained is a straight line by taking more than two points and joining them as the equation is a first degree linear equation. The complete plot of the graph using 5 points (1,5) , (0,3), (–1,1), (–2, –1), (–3, –3) has been shown below which is a straight line as expected. ## WorkSheets

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