EQUATIONS WITH INFINITELY MANY SOLUTIONS OR NO SOLUTION WORKSHEET

Problem 1 :

Solve the following equation :

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

Problem 2 :

Solve the following equation :

(1/2)(8y - 6) = 5y - (y + 3)

Problem 3 :

Solve the following equation :

(1/3)(9 - 6x) = 5 - 2x

Problem 4 :

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

If the linear equation above is an identity, what is the value of a?

Problem 5 :

In no value of x is the solution to the following equation, what is the value of a ? 

2ax - 15 = 3(x - 1) + 5(x + 2)

Answers

1. Answer :

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

Simplify both sides. 

2 - 2x + 5x = 3x + 3

2 + 3x = 3x + 3

Subtract 3x from each side. 

2 = 3

The above result is false. Because 2 is not equal to 3. Because the result we get at the last step is false, the given equation has no solution.

2. Answer :

(1/2)(8y - 6) = 5y - (y + 3)

Simplify both sides. 

4y - 3 = 5y - y - 3

4y - 3 = 4y - 3  

Subtract 4y from each side. 

-3 = -3

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.

3. Answer :

(1/3)(9 - 6x) = 5 - 2x

Simplify. 

3 - 2x = 5 - 2x 

Add 2x to each side.

3 = 5

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions. 

4. Answer :

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

Simplify. 

5 - 2x = 5 - ax  

Because the given equation is an identity, the coefficients of like terms on both sides must be equal.

That is, coefficients of 'x' terms on the left side and right side must be equal. 

So, equate the coefficients of 'x'. 

-2 = -a

Multiply each side by (-1). 

a = 2

5. Answer :

2ax - 15 = 3(x - 1) + 5(x + 2)

Simplify. 

2ax - 15 = 3x - 3 + 5x + 10

2ax - 15 = 8x + 7

From the given information, it is clear that the equation has no solution.

In the simplified equation above, if 2a = 8, then

8x - 15 = 8x + 7

Subtract 8x from both sides.

-15 = 7 (False)

In the given equation, if 2a = 8, then the equation has no solution.

2a = 8

Divide both sides by 2.

a = 4

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