SOLVING EQUATIONS WITH NO SOLUTION AND IDENTITY

It is possible that an equation may have no solution. That is, there is no value of the variable that will result in a true equation. It is also possible that an equation may be true for all values of the variable. Such an equation is called an identity.

When you are solving a linear equation in one variable, you will have the following situation.

Case (i) :

The variable vanishes and you get a false statement. Then the equation has no solution.

Case (ii) :

The variable vanishes and you get a true statement. Then the equation is identity. That is, the equation has infinitely many solutions.

Case (iii) :

The variable does not vanish. Then the equation has only one solution.

Example 1 :

Solve :

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

Solution :

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

Use the Distributive Property.

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

2 + 3x = 3x + 3

Subtract 3x from both sides.

2 = 3

The variable x vanishes in the last step and 2 = 3 is a false statement. Therefore, the given equation has no solution.

Example 2 :

Solve :

5w - 3(1 - w) = -2(3 - w)

Solution :

5w - 3(1 - w) = -2(3 - w)

Use the Distributive Property.

5w - 3 + 3w = -6 + 2w

8w - 3 = -6 + 2w

Subtract 2w from both sides.

6w - 3 = -6

Add 3 to both sides.

6w = -3

Divide both sides by 6.

w = -¹⁄₂

The variable w does not vanish in the last step. Therefore, the given equation has only one solution, that is -¹⁄₂.

Example 3 :

Solve :

(¹⁄₂)(8y - 6) = 5y - (y + 3)

Solution :

(¹⁄₂)(8y - 6) = 5y - (y + 3)

Use the Distributive Property.

(¹⁄₂)(8y) - (¹⁄₂)(6) = 5y - y - 3

4y - 3 = 4y - 3

Subtract 4y from both sides.

-3 = -3

The variable y vanishes in the last step and -3 = -3 is a true statement. Therefore, the given equation is identity, that is, the equation has infinitely many solutions.

Example 4 :

Solve :

()(9 - 6x) = 5 - 2x

Solution :

()(9 - 6x) = 5 - 2x

Use the Distributive Property.

()(9) - ()(6x) = 5 - 2x

3 - 2x = 5 - 2x

Add 2x to both sides.

3 = 5

The variable x vanishes in the last step and 3 = 5 is a false statement. Therefore, the given equation has no solution.

Example 5 :

Solve :

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

Solution :

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

Use the Distributive Property.

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

2x - 10 = 2x - 10

Subtract 2x from both sides.

-10 = -10

The variable x vanishes in the last step and -10 = -10 is a true statement. Therefore, the equation is identity, that is, the given equation has infinitely many solutions.

Example 6 :

Solve :

Solution :

-7(2x - 3) + 8x = 15 + 6x

-14x + 21 + 8x = 15 + 6x

-6x + 21 = 15 + 6x

-12x + 21 = 15

-12x = -6

x = ¹⁄₂

The variable x does not vanish in the last step. Therefore, the given equation has only one solution, that is ¹⁄₂.

Example 7 :

Solve :

Solution :

13 - 7(x + 1) = 9x - 6

13 - 7x - 7 = 9x - 6

-7x + 6 = 9x - 6

-16x + 6 = -6

-16x = -12

x = ¾

The variable x does not vanish in the last step. Therefore, the given equation has only one solution, that is ¾.

Example 8 :

Solve :

-2[3 - (x - 4)] + 5x = 7(x - 2)

Solution :

-2[3 - (x - 4)] + 5x = 7(x - 2)

-2[3 - x + 4] + 5x = 7x - 14

-2[-x + 7] + 5x = 7x - 14

2x - 14 + 5x = 7x - 14

7x - 14 = 7x - 14

-14 = -14

The variable x vanishes in the last step and -14 = -14 is a true statement. Therefore, the given equation is identity, that is, the equation has infinitely many solutions.

Example 9 :

Solve :

0.4(5x - 9) = 6x + 4(0.3 - x)

Solution :

0.4(5x - 9) = 6x + 4(0.3 - x)

2x - 3.6 = 6x + 1.2 + 4x

2x - 3.6 = 2x + 1.2

-3.6 = 1.2

The variable x vanishes in the last step and -3.6 = 1.2 is a false statement. Therefore, the given equation has no solution.

Example 10 :

()(25 - 10x) = 5 - ax

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

Solution :

()(25 - 10x) = 5 - ax

5 - 2x = 5 - ax

If a = 2,

5 - 2x = 5 - 2x

5 = 5

The variable x vanishes in the last step and 5 = 5 is a true statement. Then the given equation is identity.

If a = 2, the given equation is identity.

Example 11 :

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

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

Solution :

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

4x + 13 = 7x - 14 + bx

4x + 13 = bx + 7x - 14

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

If b + 7 = 4,

4x + 13 = 4x - 14

13 = -14

The variable x vanishes in the last step and 13 = -14 is a false statement. Then the given equation has no solution.

b + 7 = 4

b = -3

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

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