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Linear Equations

Last Updated : 23 May, 2024
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Linear equation is an equation in which the highest power of the variable is always 1. Linear equations are also called on-degree equations. These equations represent a line in the coordinate geometry. They can have one, two, three, or more variables. The standard form of the linear equation is Ax + By = 0.

Linear-Equations

Linear Equations

In this article, we will learn about Linear equations, their types, and examples in detail.

Linear Equation Definition

A linear equation as the name suggests is an equation where the degree of the equation is one, i.e. it has the highest power of the variable as one.

Any equation is an algebraic expression with a (=) sign and if that expression is linear then the formed equation is also linear. A linear equation can have one, two, three, or more variables and these represent the straight lines.

Standard-Form-of-Linear-Equation

Linear Equation

Examples of Linear Equations

Some examples of linear equations are,

  • x – 11 = 0 (Linear Equation with One Variable)
  • x + y = 7 (Linear Equation with Two Variables)
  • x – y + z = 12 (Linear Equation with Three Variables)

A linear equation can have various examples and it can be categorized on the basis of the number of variables it holds.

Let’s learn the most common linear equations in the table below.

Linear Equations with One Variable Linear Equations with Two Variables Linear Equations with Three Variables
Linear equations, i.e. the equation with power one and with one variable are called the linear equation with one variable. Linear equations, i.e. the equation with power one and with two variables are called the linear equation with two variables. Linear equations, i.e. the equation with power one and with three variables are called the linear equation with three variables.
To find the unique solution to these equations only one equation is required. To find the unique solution to these equations two equations are required. To find the unique solution to these equations three equations are required.
Example: x + 4 = 6 Example: x + + y + 4 = 6 Example: x + y + z + 4 = 6

Linear Equation Formula

The formula used to represent the linear equations is called the linear equation formula. There are various ways to represent the linear equations such as,

Equation Form

Equation

Example

Standard Form

ax + by = c

2x + 3y = 6

Slope-Intercept Form

y = mx + b

y = 2x + 3

Point-Slope Form

y – y₁ = m(x – x₁)

y – 4 = 3(x – 2)

Intercept Form

x/a + y/b = 1

x/2 + y/3 = 1

Standard Form of Linear Equation

Standard form of the linear equation in two variables is,

Ax + By = C

Where A and B are never zero.

Some examples of the linear equation in two variables in standard form are,

  • 2x + 3y = 11
  • 5x – y = 6

Slope Intercept Form

Slope-Intercept form of the linear equation in two variables is,

y = mx + c

where 

  • m is the slope of the line, and
  • c is the intercept on the y-axis.

Some examples of the linear equation in slope-intercept in two variables are,

  • y = 3x + 4
  • y = 11/3x – 3

Point Slope Form

Point-Slope form of the linear equation in two variables is,

y – y1 = m(x – x1)

Where,

  • m is the slope of the line, and
  • (x1, y1) is the given point.

Some examples of the linear equation in slope-intercept in two variables are,

  • y – 11 = 3(x – 6)
  • y + 5 = 4(x + 11)

Intercept Form

Intercept form of the linear equation in two variables is,

x/a + y/b = 1

Where a and b are the x and y intercepts cut by the graph of the line.

Some examples of the linear equation in intercept form for two variables are,

  • x/2 + y/3 = 1
  • x/3 – y/2 = 1
  • y/7 – x/4 = 1

How to Solve Linear Equations?

The solution to the linear equation is mostly studied under two heading

  • Solution of Linear Equations in One Variable
  • Solution of Linear Equations in Two Variable

Let’s learn about both in detail.

Solution of Linear Equations in One Variable

To solve the linear equation in one variable we first isolate the linear equation where all the variables are on the LHS and the constants are on the RHS and each side is individually simplified and then solved to get the required solution. This can be understood with the help of the example discussed below.

Example: Solve the equation, 4x – 6 = 2 + 2x

Solution:

Given,

4x – 6 = 2 + 2x

Taking all the variables on the LHS and the constants on the RHS

4x – 2x = 2 + 6

⇒ 2x = 8

⇒ x = 4

This is the required solution of the given equation.

Solution of Linear Equations in Two Variables

To solve the linear equation in two variables we need two linear equations that are solved to get the solution of the linear equation. There are various methods to solve the linear equations in two variables such as Graphical Method, Simultaneous Solving, Substitution Methods, etc. This can be understood by the example discussed below,

Example: Solve the equation, x + y = 12 and y = x – 2

Solution:

Given Equation,

  • x + y = 12…(i)
  • y = x – 2…(ii)

putting the value of y from eq (ii) in eq (i) we get

x + (x – 2) = 12

⇒ x + x – 2 = 12

⇒ 2x = 12 + 2

⇒ 2x = 14

⇒ x = 7

Putting the value of x in eq (i)

7 + y = 12

⇒ y = 12 – 7 = 5

Thus,

  • x = 7
  • y = 5

Linear Equation Graph

The graph of the linear equation generally represents a straight line. The linear equation in one variable represents a straight line parallel to either axis, this can be understood as,

x + 7 = 0

This linear equation in one variable represents a straight line passing through the point (-7, 0) and parallel to the y-axis. Similarly linear equations in two variables also represent a straight line and its graph can be plotted by following the steps discussed below,

Example: Plot the graph for a linear equation in two variables, x  + y –  6 = 0

Use the following steps to plot the graphs

Step 1: Arrange the given equation of the line in the standard form as, x + y = 6

Step 2: Now change the equation in the intercept form by dividing 6 on both sides to make the RHS 1. 

x/6 + y/6 = 1

Step 3: The denominator of x and y represents the intercept on the x and y axis respectively. The intercept on the x-axis is 6 and the intercept on the y-axis is 6.

Step 4: Find the point on the x-axis and the y-axis, i.e. the point on the x-axis is (6, 0) and the point on the y-axis is (0, 6). Join these points to get the line.

Graph of Linear Equation

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Solving Linear Equations

Example 1: Solve 2x  = 3(x + 4)

Solution:

Given equation,

2x  = 3(x  + 4)

⇒ 2x = 3x + 3(4)

⇒ 2x = 3x + 12

⇒ 2x – 3x = 12

⇒ -x = 12

⇒ x = -12

Example 2: Solve 2x – y = 4 and x + y = 5

Solution: 

Given Equation,

  • 2x – y = 4…(i)
  • x + y = 5…(ii)

From eq. (ii) y = 5 – x

Putting the value of y from eq (ii) in eq (i) we get

2x + (5 – x) = 4

⇒ 2x + 5 – x = 4

⇒ 2x – x = 4 – 5

⇒ x = -1

Putting value of x in eq (i)

2(-1) – y = 4

⇒ -2 – y = 4

⇒ y = -2 – 4

⇒ y = -6

Thus,

  • x = -1
  • y = -6

Example 3: Solve x – 7 = 2(x – 3)

Solution:

Given equation,

x – 7  = 2(x -3)

⇒ x – 7 = 2x – 6

⇒ -7 + 6 = 2x – x

⇒ x = -1

Example 4: Solve 2x + 3y = 6 and x – y = 3.

Solution: 

Given Equation,

  • 2x + 3y = 6 . . .(i)
  • x – y = 3  . . .(ii)

take equation (ii), 

x = y + 3 . . .(iii)

putting the value of xfrom eq (iii) in eq (i) we get

2(y + 3) + 3y = 6

⇒ 2y + 6 + 3y = 6

⇒ 5y = 6 – 6

⇒ 5y = 0

⇒ y = 0

Putting the value of y in eq (iii)

x = y + 3 = 0 + 3 = 3

  • x = 3
  • y = 0

FAQs on Linear Equations

What is Linear Equation?

Linear equations are the equation, with degree one i.e. in these equations the highest power of the variable is one. These equations can have one, two, three, or more variables.

What is linear equation and simple equation?

Linear equations are euqtaions with highest power of variable to be 1 and linear equation in one variable is called simple equation.

What are Examples of Linear Equations?

Some examples of linear equations are,

  • 2x – 1 = 0 (Linear Equation with One Variable)
  • x + 3y = 17 (Linear Equation with Two Variables)
  • x – y + 2z = 2 (Linear Equation with Three Variables)

What is the Formula for Linear Equation?

Formula for the linear equation is,

In standard form,

  • ax + by + c = 0

In slope form,

  • y = mx + c

How To Solve Linear Equations?

Linear equations can be solved by various methods we can solve linear equations in one variable, linear equations in two variables, etc by different methods.

For a linear equation with two variables, we required two equations, that can be solved by various methods such as substitution method, simultaneous equation solving, etc.

Can a Linear Equation in Two Variables have More Than One Solution?

Yes, a linear equation in two variables can have more than one solution, as when a graph of both linear equations coincides, then it can have infinitely many solutions.

Can Linear Equation have More Than Two Variables?

Yes, a linear equation can have however many multiple variables as long as all the variables have at most a degree of 1. 



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