Solution of a Linear Equation

In Mathematics, a linear equation is defined as an equation that is written in the form of Ax+By=C. It is the combination of two variables and a constant value present in them. When solving the system of linear equations, we will get the values of the variable, which is called the solution of a linear equation. In this article, we will learn what is the solution of a linear equation, the types of solutions of a linear equation and so on.

Table of Contents:

What is Meant by Solution of a Linear Equation?

The solution of a linear equation is defined as the points, in which the lines represent the intersection of two linear equations. In other words, the solution set of the system of linear equations is the set of all possible values to the variables that satisfies the given linear equation.

Types of Solutions for Linear Equations

There are 3 possible types of solutions to the set of linear equations. They are:

  • Unique Solution
  • No Solution
  • Infinitely Many Solutions

Now, let us discuss all the types of solutions for linear equations in detail.

Unique Solution

The linear equation in one variable always has a unique solution. The unique solution of a linear equation represents that there exists only one point, which on substitution, L.H.S becomes equal to R.H.S. In the case of simultaneous linear equations in two variables, the solution should be an ordered pair (x, y). In this case, the ordered pair will satisfy the set of equations.

No Solution

If the graphs of the linear equations are parallel, then the system of linear equations has no solution. In this case, there exists no point such that no lines intersect each other.

Infinitely Many Solutions

A linear equation in two variables has infinitely many solutions. For the system of linear equations, there exists a solution set of infinite points for which the L.H.S of an equation becomes R.H.S. The graph for the system of linear equations with infinitely many solutions is a graph of straight lines that overlaps each other.

Finding the solution of a Linear Equation in One Variable

The solution of a linear equation in one variable can be easily found by keeping the variable on the one side and constant on the other side of an equation.

For example, 3x+2 = 11

Now, keep 3x on the left-hand side and bring 2 to the right hand side.

Thus, the given equation becomes,

3x = 11-2

3x = 9

x = 9/3

x = 3

Thus, the unique solution of the given linear equation is x = 3.

Verification:

If we substitute x = 3 in the given equation, L.H.S should be equal to R.H.S

3(3)+ 2 = 11

9+2 = 11

11 = 11

Hence, the solution x = 3 satisfies the given equation.

Finding the solution of a Linear Equation in Two Variables

The solution of a linear equation in two variables can be found using different methods, such as:

Note:

  1. An independent system has exactly one solution pair. (A solution should be a point where two lines intersect)
  2. A dependent system has infinitely many solutions (The line coincides each other and they are the same line)
  3. An inconsistent system has no solution. (The lines are parallel and they do not intersect at any point)

Solved Examples

Example 1:

Find two solutions for the equation: 4x+3y = 12

Solution:

Given equation: 4x+3y= 12

Take x = 0, then we get

4(0) + 3y = 12

3y=12

y=12/3 = 4

Thus, (0, 4) is one solution.

Similarly, take y=0, then we get

4x + 3(0) = 12

4x = 12

x = 12/4

x = 3.

Thus (3, 0) is another solution.

Thus, the two solutions for the equations 4x+3y=12 are (0, 4) and (3, 0).

Example 2:

How many Solutions do the equations -2x+y=9 and -4x+2y=5 have?

Solution:

The equations -2x+y=9 and -4x+2y=5 have no solution.

The line equations -2x+y=9 and -4x+2y=5 are parallel to each other, and hence, they do not have solutions.

Practice Problems

Solve the following problems:

  1. Find two solutions for the equation: 2x+5y = 0.
  2. Determine the value of k, if a = 2 and b=1 is the solution of the equation 2a+3b=k.

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Frequently Asked Questions on Solution of a Linear Equation

What is meant by the solution of a linear equation?

A solution of a linear equation is the set of all possible values of a variable, which should satisfy the given equations.

How many types of solutions are possible for a system of linear equations?

The system of linear equations can have three types of solutions. They are:
Unique Solution
No solution
Infinitely many solutions.

What are the three methods which can be used to solve the linear equations in two variables?

The most common three methods used to solve linear equations in two variables are:
Substitution Method
Elimination Method
Graphing Method

What is the graph of the linear equations if they have no solutions?

If the linear equation has no solution, then the graph of the linear equation should be parallel to each other and they never intersect.

Mention the types of linear systems.

The types of linear systems are:
Dependent system
Independent system
Inconsistent system

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