EQUATIONS WITH MANY SOLUTIONS OR NO SOLUTION

When we solve a linear equation in one variable, we may find exactly one value of x that will make the equation a true statement. But, when we simplify some equations, we may find that they have more than one solution or they do not have solution.

The table given below will help us to find the number of solutions to a linear equation in one variable.

Examples

Example 1 :

In the linear equation given below, say whether the equation has exactly one solution or infinitely many solution or no solution. 

4x - 3  =  2x + 13

Solution : 

4x - 3  =  2x + 13

Add 3 to both sides. 

4x  =  2x + 16

Subtract 2x from each side. 

2x  =  16

Divide each side by 2. 

x  =  8

Justify and Evaluate :

Substitute x  =  8 in the given equation. 

4(8) - 3  =  2(8) + 13  ?

32 - 3  =  16 + 13  ?

29  =  29 ------> True

Substitute some other value for x, say x  =  10. 

4(10) - 3  =  2(10) + 13  ?

40 - 3  =  20 + 13  ?

37  =  23  False

Only x  =  8 makes the equation a true statement and not any other value.

So, there is only one solution, that is x  =  8.  a  

Example 2 :

In the linear equation given below, say whether the equation has exactly one solution or infinitely many solution or no solution. 

4x - 5  =  2(2x - 1) - 3

Solution : 

4x - 5  =  2(2x - 1) - 3

Use distributive property.

4x - 5  =  2(2x) - 2(1) - 3

Simplify

4x - 5  =  4x - 2 - 3

4x - 5  =  4x - 5

We find the same coefficient for x on both sides.

So, subtract 4x on both sides to get rid of x-terms. 

-5  =  -5

When we solve the given equation, we don't find 'x' in the result.

But the statement (-5  =  -5) we get at last is true.

So there are infinitely many solutions.

Example 3 :

In the linear equation given below, say whether the equation has exactly one solution or infinitely many solution or no solution. 

4x + 2  =  4x - 5

Solution : 

Solve the given equation.

We find the same coefficient for x on both sides.

So, subtract 4x on both sides to get rid of x-terms. 

4x + 2  =  4x - 5

2  =  -5

When we solve the given equation, we don't find "x" in the result.

But the statement (2  =  -5) we get at last is false.

So, there is no solution. 

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