EQUATIONS WITH MANY SOLUTIONS OR NO SOLUTION WORKSHEET

Problems 1-5 : In each of the linear equation, say whether the equation has infinitely many solutions or no solution.

Problem 1 :

4x - 3 = 2x + 13

Problem 2 :

4x - 5 = 2(2x - 1) - 3

Problem 3 :

4x + 2 = 4x - 5

Problem 4 :

5(x - 2) - 3x = 2(x - 5)

Problem 5 :

(1/2)(6 - 4x) = 5 - 2x

Problem 6 :

kx = 3x + 5

If the linear equation above, if k = 3, does it have infinitely many solutions or no solution?

Problem 7 :

(1/3)(15 - 6x) = 5 - ax

If the linear equation above has infinitely many solutions, what is the value of a?

Problem 8 :

4x + 13 = 7(x - 2) + bx

If the linear equation above has no solution, which of the following could be the value of b?

Answers

1. Answer :

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, the given equation has 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 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, the given equation equation has infinitely many solutions.

3. Answer :

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, the given equation has no solution.

4. Answer :

5(x - 2) - 3x = 2(x - 5)

Use distributive property.

5x - 10 - 3x = 2x - 10

Simplify

5x - 10 - 3x = 2x - 10

2x - 10 = 2x - 10

We find the same coefficient for x on both sides.

So, add 2x to both sides to get rid of x-terms. 

-10 = -10

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

But the statement (-10 = -10) we get at last is true.

So, the given equation has infinitely many solutions.

5. Answer :

(1/2)(6 - 4x) = 5 - 2x

Use distributive property.

(1/2)(6) - (1/2)(4x) = 5 - 2x

Simplify

3 - 2x = 5 - 2x

We find the same coefficient for x on both sides.

So, add 2x to both sides to get rid of x-terms. 

3 = 5

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

But the statement (3 = 5) we get at last is false.

So, the given equation has no solution.

6. Answer :

kx = 3x + 5

Given : k = 3.

3x = 3x + 5

Subtract 3x from both sides.

0 = 5

(false statement)

So, the given equation has no solution when b + 7 = 4.

b + 7 = 4

Subtract 7 from both sides.

b = -3

Therefore, the given equation has no solution, if k = 3.

7. Answer :

(1/3)(15 - 6x) = 5 - ax

Use Distributive Property.

(1/3)(15) - (1/3)(6x) = 5 - ax

5 - 2x = 5 - ax

Subtract 5 from both sides.

-2x = -ax

Multiply both sides by -1.

2x = ax

In the equation above, if a = 2,

2x = 2x

The above equation is true for all real values of x. That is, the above equation has infinitely many solutions.

Therefore, the given equation has infinitely many solutions when a = 2.

8. Answer :

4x + 13 = 7(x - 2) + bx

Use Distributive Property.

4x + 13 = 7x - 14 + bx

4x + 13 = bx + 7x - 14

4x + 13 = (b + 7)x - 14

In the equation above, if b + 7 = 4,

4x + 13 = 4x - 14

13 = - 14

(false statement)

So, the given equation has no solution when b + 7 = 4.

b + 7 = 4

Subtract 7 from both sides.

b = -3

Therefore, the given equation has no solution when b = -3.

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