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 :
(⅓)(9 - 6x) = 5 - 2x
Solution :
(⅓)(9 - 6x) = 5 - 2x
Use the Distributive Property.
(⅓)(9) - (⅓)(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 :
(⅕)(25 - 10x) = 5 - ax
If the linear equation above is an identity, what is the value of a?
Solution :
(⅕)(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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