In mathematics, linear equations are very important to know for solving the further algebra problems. It has various uses in day-to-day life also. Even the statisticians use linear and nonlinear equations to predict what will happen in future.

linear-Equations

What is an Equation?

Before knowing linear equations let's know what is an expression and equation?

Expression: An expression can be a combination of numbers, combination of variables, a single number, or a single variable and operational symbols.

For Example,  X + 1, 2•1 + 1, X ÷ 15

These are the perfect examples of an expression as there is no equal to sign in between.

Equation: An equation can be defined as a statement that two quantities are equivalent.

For Example, X + 1 = 7

Here the left hand side (X+1)  is equal to the right hand side (7) which is an equation. Equations must have equals to (=) sign in between the expressions.


What is a Linear Equation?

A linear equation is an equation in which the highest degree or power of each veritable is one and can never be more than one and which describes a straight line on a graph.


Linear Equation Examples

To make the linear equations you can simply use the standard form explained above. Instead of A, B and C you need to put numbers. For example,

  • 3x + 4y = 5
  • x + 2y = 3
  • 2x + y = 9
  • 3x = 6
  • y/2 = 4

There are any exponents (power), square roots and cube roots are found with variables (like x or y) then it will not be a linear equation. Here are some nonlinear equations examples,

  • 2√x − y = 4
  • y2(square) − 4 = 2
  • x3(cube)/2 = 8

Also Read: 

Forms of Linear Equation

How to identify that any equation is a linear equation? Well, there are some forms of linear equations so that you can easily find out the equation is a linear equation or nonlinear equation.

To write a linear equation you can use these forms of equations explained below.

1. General form

This form of linear equation represents a straight line. The general form of linear equation can be written as,

Ax + By + C = 0                                       [ Graph ]

Where A and B can be zero (0), but at the same time they can not be zero.

The general form is useful to make equations for vertical lines which is not possible by point slope form and slope intercept form. For example, 2x + 3 = 0.

2. Standard form

Standard form of linear equation is used to find parallel and perpendicular lines. This form of equation can be written as,

Ax + By = C                                         [ Graph ]

This is the standard form of linear equation where A, B and C are coefficients (numbers) while x and y are variables. A and B can be zero but not at the same time.

3. Slope Intercept form

The form is useful to draw a line on a graph. Only the slope and y-intercept line can be drawn by slope intercept form. The slope intercept form can be written as,

y = Mx + B                                           [ Graph ]

Where, m is the slope of a line and b is the y-intercept. Vertical lines and having unidentified slope can not be drawn by this form of equation.

4. Point Slope form

The form is used when there is a point given on a line and the slop. And by using these two points we need to find the equation of the line.

Point slope form of linear equation can be written as,

y - y1 = m(x - x1)                                      [ Graph ]

Where, the m is a slope of the line and (x1,y1) represents any point on the line.

5. Vertical form

Vertical form of linear equation can be drawn as a straight line on graph vertically on x-intercept.

Vertical form of linear equation can be written as,

x = a                                                         [ Graph ]

Where, a is coefficient (number) and x is the variable.

6. Horizontal form

Horizontal form of linear equation is also a straight line which can be drawn horizontally on y-intercept.

Horizontal form of linear equation can be written as,

y = b.                                                        [ Graph ]

Where, b is a coefficient and y is a variable in the equation.


Formula and Methods to solve linear equations

There are mainly three direct methods available by which we can easily solve any linear equations.

Matrix method and interactive methods are also used when the problem is big and completed. Here we are going to discuss only the direct methods to solve linear equations.

  1. Substitution method
  2. Elimination method
  3. Cross-Multiplication method


Graphing Linear Equations

The graphical representation of a linear equation is a straight line (that's why they call it linear). To draw a graph linear equation you can put the numbers for x and y on any linear equation then we will get two intercept points.

The intercept points are when x = 0 or y = 0. There are some steps to follow to graph a linear equation:

  • Put x = 0 into the equation and solve for y
  • Now you have point (0,y) on y-axis
  • Put y = 0 into the equation and solve for x
  • Now you have point (x,0) on the x-axis
  • Now you have two points and you can draw a straight line by the help of two points.

You can also try this equation with the other numbers and you will get the same answer which represents a straight line on a graph.

Example Problem:

Graph the linear equation: 2x + y = 2

As you know the steps to graph a linear equation we will follow the steps one by one.

Step 1: Put x = 0 and solve for y.

2x + y = 2
2(0) + y = 2
y = 2

Step 2: Put y = 0 and solve for x.

2x + y = 2
2x + 0 = 2
x = 1

Step 3: Now we have two intercept points (0,2) and (1,0)

Step 4: Draw a straight line with the help of the two points.

[ Graph ]

You can check your answer by trying other numbers in the equation.

Example : Let's put 2 for x and solve

2x + y = 2
2(2) + y = 2
y = 2 - 4
y = -2

The point we get (2,-2).

Now you will find that the point is on your line which means your answer is correct. You can also try some other numbers to double check as well.