Linear equations are equations containing variables which have an exponent of either zero or one. When we solve linear equations it can give us three different types of solutions. The answers could be either one solution or infinite solution or no solution. One solution mean the variable has a single value infinitely many solutions mean when we plug in any value of the variable it suffices the equation and linear equations with no solution mean the variable does not carry any value.

Linear equations with no solution are defined as when we try to solve linear equations we do not get any answer to the equation. Whichever value of the variable be substituted it is impossible for the equation to be true.

Suppose there are two straight lines having the same slope but not part of same straight line. This could be in case of two parallel lines which never intersects with each other. There is no common point $(x, y)$ which satisfies the equation of each straight line. Such lines are known as inconsistent lines, and there is no solution to it. It cannot be understood that the slope $f$ the two lines are equal as they are not written in one $f$ the standard forms of straight lines. When we try to solve these equations algebraically, we assume the equations to be true which is contradictory to the fact. Actually we are working on false assumption which is proved at the end of solving and then we call the equations inconsistent or no solution.

Two linear equations with no solution, whereby there are two equations given which on trying to solve gives us no solution.

Two linear equations with no solution, whereby there are two equations given which on trying to solve gives us no solution.

Few examples on linear equations with no solutions are illustrated below:

**Example 1:** Solve the following equation:

3(3 + 3x) -16x = 7(4x - 5x)

3(3 + 3x}-16x = 7(4x - 5x)

We see that there is existence of variables on both sides of the equal to. Same goes with the numbers as they are present on either side of the equal to.

9 + 9x - 16x = -7x

9 - 7x = -7x

9 = - 7x + 7x

9 = 0

9 = 0

Here the terms in $x$ cancels each other. We see that when we subtract $7x$ from $7x$ it gives us $0$. As a result there are no terms in $x$ left in the equation to solve. Hence, no solution

Example 2

$\frac{1}{2}$ $(2 - 4x)$ = 13 - 2x

$\frac{1}{2}$ $(2 - 4x)$ = 13 - 2x

We see that there is existence of variables on both sides of the equal to. Same goes with the numbers as they are present on either side of the equal to.

1/2 $\times$ (2) - 1/2 $\times$ (4x) = 13 - 2x

1 - 2x = 13 - 2x

2x - 2x = 13 - 1

0 = 12

Here the terms in $x$ cancels each other. We see that when we subtract $2x$ from $2x$ it gives us $0$. As a result there are no terms in $x$ left in the equation to solve. Hence, no solution

Example 3

4x - 3y = 8

8x = 2 (7 + 3y)

4x - 3y = 8 ...(1)

8x = 2 (7 + 3y) ...(2)

8x = $2 \times 7 + 2 \times 3y$

8x = 14 + 6y

8x = 14 + 6y

8x - 6y = 14

-2 $\times$ (4x - 3y = 8)

-8x + 6y = 16

8x - 6y = 14

-8x + 6y = 16

_______________

0 x + 0 y = 30

_______________

This implies: 0 = 30

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This implies: 0 = 30

Solve below listed problems to sharp your skills:

**Problem 1**: 7x - 8y = 12 and 7x = 2 (6 + 4y)