Linear Equations with No Solutions

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.

What Does it mean when an Equation has No Solution?

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.

System of Linear Equations with No Solution

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.

No Solution Graph

Graph of Linear Equations with No Solution: Linear equations with no solutions are parallel lines with no point of intersection. There is no common point lying between the pair of straight lines.

Examples

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

Example 1: Solve the following equation:

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

Solution:

Step 1: Write down the equation given

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.

Step 2: We need to separate the terms containing variables on one side and the constants on the other side of the equal to. Combining the like terms shall give us

9 + 9x - 16x = -7x

9 - 7x = -7x
9        = - 7x + 7x
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

Step 4: As there is no solution obtained in the given linear equation, we can conclude that the equations are inconsistent in form.

Example 2
:  Solve the following equation:

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

Solution:

Step 1: Write down the equation given

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

Step 2: Using distributive property on left side of the equation.

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

Step 3: We need to separate the terms containing variables on one side and the constants on the other side of the equal to. Combining the like terms shall give us

2x - 2x = 13 - 1

Step 4: Solving either side of the equation. So, it becomes

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

Step 5: As there is no solution obtained in the given linear equation, we can conclude that the equations are inconsistent in form.

Example 3
:  Solve two linear equations:
4x - 3y = 8
8x = 2 (7 + 3y)

Step 1: Let us name the first equation as equation number $1$ and the second equation as equation number $2$.
4x - 3y = 8   ...(1)
8x = 2 (7 + 3y)  ...(2)
Step 2: Parenthesis is present on the right side of the second equation. Using distributive property to expand it

8x = $2 \times 7 + 2 \times 3y$

8x = 14 + 6y

Step 3: The equation number $1$ has its terms containing variables on one side. Similarly, we need to get the terms containing variables on one side of the second equation

8x = 14 + 6y

8x - 6y = 14

Step 4: Using the method of elimination to solve for the variables $x$ and $y$. We have to get the coefficient of either $x$ or $y$ same with negative sign for both the equations. So, we multiply the equation number $1$ with $-2$ throughout.

-2 $\times$ (4x - 3y = 8)

-8x + 6y = 16

Step 5: Now we add the two equations

8x - 6y = 14

-8x + 6y = 16
_______________
0 x + 0 y = 30
_______________

This implies: 0 = 30

Step 6: This results in cancellation of both the variables giving us $0$. Hence, no solution

Step 7: As there is no solution obtained in the given pair of equation, we can conclude that the equations are inconsistent in form.

Practice Problems

Solve below listed problems to sharp your skills:
Problem 1: 7x - 8y = 12  and 7x = 2 (6 + 4y)
Problem 2: Find the solution of the linear equation: x + 5x + 4 = 3 (2x - 1)
Problem 3: Solve 10x + 15x - 4 = 3(2x - 3) + x