DETERMINING THE NUMBER OF SOLUTIONS

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 explains the situations where we have exactly one solution, more than one solution and infinitely many solutions for a linear equation in one variable. 

Example 1 :

Use the properties of equality to simplify the equation given below. Say whether the equation has one, zero, or infinitely many solutions.

4x - 3  =  2x + 13

Solution : 

4x - 3  =  2x + 13

Subtract 3x from each side.

2x - 3  =  13

Add 3 to 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.

Example 2 :

Use the properties of equality to simplify the equation given below. Say whether the equation has one, zero, or infinitely many solutions.

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 from each side 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 :

Use the properties of equality to simplify the equation given below. Say whether the equation has one, zero, or infinitely many solutions.

4x + 2  =  4x - 5

Solution : 

We find the same coefficient for x on both sides.

So, subtract 4x from each side to get rid of x-terms. 

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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