# Linear Equations

## Definition of Linear Equations

The equations which are of first order and first degree are called linear equations. The Linear Equations represents straight lines in the $x-y$ Cartesian Coordinate system.

The linear equations with $n$ number of terms can be written in the following form:

${a_1}{x_1} + {a_2}{x_2} + {a_3}{x_3} + …………. + {a_n}{x_n} + b = 0$

Where, ${x_1},\,\,{x_2},\,\,{x_3},\,\,………..,\,\,{x_n}$ are the variables or unknowns whereas $b,\,\,{a_1},\,\,{a_2},\,………….\,\,,\,\,{a_n}$ are coefficients.

## Example of Linear equations

To understand the concerpt of Linear Equations, let us consider the following examples:

• $4x + 5 = 0$

The above equation has only one variable i.e., $x$ and the degree of the equation is 1. Therefore, the above equation is a linear equation. The graph of the above linear equations will be a straight line parallel to the $y-axis$.

• $2x + y= 9$

In the above equation, there are two numbers of variables i.e., $x$ and $y$. However, the degree of the above equation is 1 since 1 is the highest power of the variables. Therefore, the above equation is a linear equation.

• $4{x^2} + 2 = 0$

The equation in the above example has only one variable i.e., $x$. However, the degree of the equation is $2$ because the highest power of the variable $x$ is $2$. Therefore, the above equation is nonlinear.

## Graph of Linear Equations

The graph of a linear equation is always a straight line when plotted in the $x-y$ Cartesian Coordinate system.

The graph of a linear equation having only one variable $x$ forms a straight line parallel to $y-axis$ and the linear equation having only one variable $y$ forms a straight line parallel to $x-axis$.

However, the graph of a linear equation having two variables is also a straight line but the line cuts both the $x$ and $y$ axis at some particular points. The point where the line cuts the $x-axis$ is called the $x-intercept$ and the the point where the line cuts the $y-axis$ is called the $y-intercept$.

### Example of Graph of a Linear Equation:

To understand the graph of a linear equation, let us take an example of linear equations as follows:

• $x+3=0$

$x=-3$

This means that for all the values of $y$, $x$ remains constant as $-3$. The graph of the above equation is a straight line parallel to $y-axis$ as shown in the figure.

• $x+y=2$

The above linear equation has two variables $x$ and $y$.

For the above equation if,

1. $x=0$, $y=2$
2. $x=1$, $y=1$
3. $x=2$, $y=0$
4. $x=3$, $y=-1$

The graph of the above equation is a straight line that passes through the $x-axis$ and $y-axis$ at the points $2$ and $2$ respectively.

## Equation of a Line

A linear equation is also called the equation of a line as they form straight lines when plotted in the $x-y$ Cartesian Coordinate system. The equation of a line or the linear equation can be expressed in different forms as detailed below:

### Equation of a line in Standard form or General Form

The standard form or general form of a linear equation is given by,

$Ax+By+C=0$

Where, $x$ and $y$ are the variables, $A$ and $B$ are the coefficients of $x$ and $y$ respectively, and $C$ is a constant.

#### Example of Equation of a line in Standard form or General Form

Some examples of equation of a line in standard form or general form are given by:

• $4x+3y+2=0$
• $2x+y-4=0$
• $-x+2y=-1$

### Equation of a Line in Slope-Intercept form

The equation of a line in slope-intercept form can be written as:

$y = mx + {y_0}$

Where, $m$ is the slope of the line and ${y_0}$ is the $y-intercept$ or the point where the line cuts the $y-axis$.

The slope $m$ measures the steep of the line with respect to the horizontal axis $x$. If $({x_1},{y_1})$ and $({x_2},{y_2})$ are any two points in the line, then, the slope $m$ can be calculated as:

$m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$

#### Example of Equation of a line in slope-intercept form

Let, $y=2x+4$ is a linear equation in slope-intercept form. Here, the slope $m=2$ and the $y-intercept$ ${y_0}=4$.

For the above equation, if,

$x=0$, $y=4$

$x=1$, $y=6$

$x=2$, $y=8$

$x=3$, $y=10$

If we plot the above values in the $x-y$ Cartesian Coordinate system then the resulting graph is as shown in the figure. In the graph, the line cuts the $y-axis$ at $y=4$. Therefore, $4$ is called $y-intercept$.

As calculated above, let any two points on the line be $(0,4)$ and $(1,6)$. Then the slope $m$ can be calculated as:

$m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$

$m = \frac{{6 - 4}}{{1 - 0}}$

$m = 2$

The equation of a line can also be defined by its $x-intercept$ ${x_0}$, and the equation can be written as:

$y = m(x - {x_0})$

$y = mx - m{x_0}$

The $x-intercept$ ${x_0}$ of the line can be found out by putting $y=0$ in the equation as shown below:

$y = 2x + 4$

$2x + 4 = 0$

$x = - 2$

The slope, $x-intercept$ and $y-intercept$ can also be found out from the general expression.

$Ax+By+C=0$

Slope, $m= - \frac{A}{B}$

$x-intercept$, ${x_0} = - \frac{C}{A}$

$y-intercept$, ${y_0} = - \frac{C}{B}$

### Equation of a Line in Point-Slope Form

If a line in the $x-y$ Cartesian Coordinate system has slope $m$ and the coordinates of a point in the line is $({x_1},{y_1})$, then, the equation of the line in Point-Slope form can be written as,

$y = {y_1} + m(x - {x_1})$

$y = {y_1} + mx - m{x_1}$

$y - {y_1} = mx - m{x_1}$

#### Example of Equation of a Line in Point-Slope Form

Let, the coordinates of a point in a line be $(2,4)$ and the slope is $3$, then,

$y - 4 = 3 \times x - 3 \times 2$

$y - 4 = 3x - 6$

$y = 3x - 6 + 4$

$y = 3x - 2$

Now, if,

$x=0$, $y=-2$

$x=1$, $y=1$

$x=2$, $y=4$

$x=3$, $y=7$

The $y-intercept$ of the line can be found out by putting $x=0$

$y=-2$

Similarly, the $x-intercept$ of the line can be found out by putting $y=0$

$3x-2=0$

$3x=2$

$x = \frac{3}{2}$

### Equation of a Line in Intercept Form

If an line having non-zero $x$ and $y-intercept$ is not parallel to any of the $x$ and $y$ axes and does not pass through the origin in the $x-y$ Cartesian Coordinate system, then, the equation of the line can be represented in Intercept Form as,

$\frac{x}{{{x_0}}} + \frac{y}{{{y_0}}} = 1$

Where, ${x_0}$ and ${y_0}$ are $x$ and $y-intercept$ respectively.

#### Example of Equation of a line in Intercept Form

Let, a line has $x-intercept$, ${x_0}=2$ and $y-intercept$, ${y_0}=4$, then the equation of the line in Intercept form will be:

$\frac{x}{2} + \frac{y}{4} = 1$

Now, if we simplify the above equation,

$\frac{{2x + y}}{4} = 1$

$2x + y = 4$

$y = 4 - 2x$

For the above equation, if

$x=0$, $y=4$

$x=1$, $y=2$

$x=2$, $y=0$

$x=3$, $y=-2$

The graph obtained by plotting the above values in the $x-y$ Cartesian Coordinate system is shown in the figure. From the figure, it is clear that the $x-intercept$ is $2$ and $y-intercept$ is $4$.

### Equation of a Line in Two Point Form

The equation of a line in Point-Slope form as discussed above is given by:

$y - {y_1} = m(x - {x_1})$

Now, if two points of the line are $({x_1},{y_1})$ and $({x_2},{y_2})$ are known, then,

$m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$

Now, the above equation can be written as,

$y - {y_1} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right)$

$\left( {y - {y_1}} \right)\left( {{x_2} - {x_1}} \right) = \left( {{y_2} - {y_1}} \right)\left( {x - {x_1}} \right)$

$\left( {y - {y_1}} \right)\left( {{x_2} - {x_1}} \right) - \left( {{y_2} - {y_1}} \right)\left( {x - {x_1}} \right) = 0$

$y{x_2} - y{x_1} - {y_1}{x_2} + {y_1}{x_1} - {y_2}x + {y_2}{x_1} + {y_1}x - {y_1}{x_1} = 0$

$\left( {{y_1} - {y_2}} \right)x + \left( {{x_2} - {x_1}} \right)y + {x_1}{y_2} - {x_2}{y_1} = 0$

#### Example of Equation of a line in Two Point Form

Let, the two points in a line be $(3, 2)$ and $(4, 4)$.

Here, ${{x_1}}=3$, ${{x_2}}=4$, ${{y_1}}=2$ and ${{y_2}}=4$.

$\left( {{y_1} - {y_2}} \right)x + \left( {{x_2} - {x_1}} \right)y + {x_1}{y_2} - {x_2}{y_1} = 0$

$(2 - 4)x + (4 - 3)y + (3 \times 4 - 4 \times 2) = 0$

$- 2x + y + 4 = 0$

$y = 2x - 4$

In the above equation, if,

$x=0$, $y=-4$

$x=1$, $y=-2$

$x-2$, $y=0$

$x=3$, $y=2$

The Graph of the above equation is shown in the figure.

### Equation of a line in Determinant form

The equation of a line in Two Point Form is given as,

$\left( {{y_1} - {y_2}} \right)x + \left( {{x_2} - {x_1}} \right)y + {x_1}{y_2} - {x_2}{y_1} = 0$

The above equation can be written in determinant form as,

#### Example of Equation of a line in determinant form

If $(1, 2)$ and $(4, 6)$ are the two points in a line, then, in determinant form, the equation of the line can be written as:

## Linear Equation as a Function

The linear equations can also be written as a function $f(x)$ or $f(y)$.

For example, let a linear equation be, $y=4x+2$

The same can also be written as $f(x)=4x+2$

### Identity Function

The linear function which when plotted in the $x-y$ Cartesian Coordinate System, passes through the origin and makes an angle of ${45^0}$ with $x$ or $y$ axis is called Identity Function.

The general representation of an Identity function is given by:

$f(x)=x$ or $f(y)=y$

#### Graph of an Identity Function

The graph of an Identity function is a straight line passing through the origin as shown in the figure.

### Constant Function

The linear function when plotted in the $x-y$ Cartesian Coordinate System forms a straight line parallel to either $x-axis$ or $y-axis$ are called constant function.

The general representation of a constant function is given by:

$f(x)=c$ or $f(y)=c$

Where, $C$ is a constant.

#### Graph of a Constant Function

The graph of a constant function is a straight line parallel to either $x-axis$ or $y-axis$ as shown in the figure.

What is the formula for a Linear Equation?

The standard form or general form of a linear equation is given by, $Ax+By+C=0$
The equation of a line in slope-intercept form can be written as: $y = mx + {y_0}$

What is Linear Equation explain with example?

The linear equations are those equations that have degree 1 and forms a straight line when plotted in the x-y Cartesian Coordinate system.
For example, the equation $4x + 5 = 0$ has only one variable and the degree of the equation is 1 and when the equation is plotted in the x-y cartesian coordinate system, it forms a straight line. Therefore, the above equation is a linear equation.

What is linear equation give 5 examples?

The linear equations are those equations that have degree 1 and forms a straight line when plotted in the x-y Cartesian Coordinate system.
5 examples of Linear Equations are:
1. $2x+3=0$
2. $-x+1=2$
3. $x+y=1$
4. $x=2y$
5. $x+y+9=0$

What are 3 Linear equations?

The example of 3 linear equations are:
1. $x-2=0$
2. $x+y=2$
3. –$x-y=0$