DETERMINING NUMBER OF SOLUTIONS WORKSHEET

Use the properties of equality to simplify each equation. Tell whether the equation has one, zero, or infinitely many solutions.

1)   4x - 3  =  2x + 13

2)  4x - 5  =  2(2x - 1) - 3

3)  4x + 2  =  4x - 5

1. Answer :

4x - 3 = 2x + 13

Add 3 to both sides.

(4x - 3) + 3 = (2x + 13) + 3

4x - 3 + 3 = 2x + 13 + 3

4x = 2x + 16

Subtract 2x from both sides.

4x - 2x = (2x + 16) - 2x

2x = 2x + 16 - 2x

2x = 16

Divide both sides by 2.

2x/2 = 16/2

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.

2. Answer :

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 from both sides to get rid of x-terms.

(4x - 5) - 4x = (4x - 5) - 4x

4x - 5 - 4x = 4x - 5 - 4x

-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.

3. Answer :

4x + 2 = 4x - 5

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 = (4x - 5) - 4x

4x + 2 - 4x = 4x - 5 - 4x

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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