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.

Frequently Asked Questions (FAQ)