Lesson Objectives
• Demonstrate an understanding of how to Solve a Linear Equation in One Variable
• Learn how to label an equation as Conditional, an Identity, or a Contradiction
• Learn how to determine if an equation has No Solution
• Learn how to determine if an equation has an Infinite Number of Solutions

Equations with No Solution or an Infinite Number of Solutions

Conditional Equations

Up to this point, we have only encountered one type of equation. This equation is known as a conditional equation. This means the equation is true under certain conditions. Suppose we saw the equation:
3x - 5 = 1
This equation is a conditional equation. It is true only under the condition that x = 2:
3(2) - 5 = 1
6 - 5 = 1
1 = 1
We can see that replacing x with a 2, results in a true statement. If we replace x with any other value, the result will be false. Let’s suppose we plugged in a 5 for x:
3(5) - 5 = 1
15 - 5 = 1
10 = 1 (false!)
This is what we mean by a conditional equation. The equation is true under certain conditions only, in the case of replacing our variable with 2, the equation was true, however, in the case of replacing our variable with a 5, our equation was false.

Identity Equation

Another type of equation is known as an identity. These equations are true for all real numbers. This means we can replace our variable with any number in the real number system and get a true statement. Let’s suppose we saw:
7(x - 5) + 2 = 7x - 33
Although each side looks different, the left side will simplify to the exact same form as the right side:
7(x - 5) + 2 = 7x - 33
7x - 35 + 2 = 7x - 33
7x - 33 = 7x - 33
Since we have the same algebraic expression (7x - 33) on each side of the equation, it does not matter what we plug in for x. The same operation is done on each side, therefore, the equation will always be true.
Let's just plug in a -1 for x and see what happens:
7(-1 - 5) + 2 = 7(-1) - 33
7(-6) + 2 = -7 - 33
-42 + 2 = -40
-40 = -40
We can see that we have the same value of (-40) on each side of the equation. You can try any number you would like for x, the equation will always be true.

The last type of equation we will discuss is known as a contradiction. A contradiction will never have a solution. No matter what we replace the variable with the equation will always be false. Let's suppose we saw:
11x - 7 = 11(x + 5)
If we simplify each side we get:
11x - 7 = 11x + 55
Think about this for a moment. On each side of the equation, we have 11x or 11 times some unknown value. No matter what that value is, it will be the same on each side of the equation. On the left side, we subtract 7 away, and on the right side, we add 55. There is no number that will make this equation true. If we subtract 11x away from each side, we get a nonsensical statement:
11x - 11x - 7 = 11x - 11x + 55
-7 = 55 (false!)
When the variable drops out and we are left with a false statement, we have a contradiction. Let's take a look at a few examples.
Example 1: Determine if each equation is conditional, an identity, or a contradiction.
-7x - 3 = -31
Let's try to solve our equation:
-7x - 3 + 3 = -31 + 3
-7x = -28 $$\frac{-7}{-7}x=\frac{-28}{-7}$$ $$\require{cancel}\frac{\cancel{-7}}{\cancel{-7}}x=\frac{4\cancel{-28}}{\cancel{-7}}$$ x = 4
Check:
-7(4) - 3 = -31
-28 - 3 = -31
-31 = -31
Since we have a solution of x = 4, we can label this equation as conditional.
Example 2: Determine if each equation is conditional, an identity, or a contradiction.
2x - 3 = 2(x - 1) - 1
Let's try to solve our equation:
2x - 3 = 2x - 2 - 1
2x - 3 = 2x - 3
We can see the same algebraic expression (2x - 3) on each side. This means any real number can replace x and give us a true statement. For this equation, we can label it as an identity.
Example 3: Determine if each equation is conditional, an identity, or a contradiction.
72 - 63x = -9(7x + 12)
Let's try to solve our equation:
72 - 63x = -9(7x + 12)
72 - 63x = -63x - 108
72 - 63x + 63x = -63x + 63x - 108
72 = -108 (false!)
We can see that we end up with a false statement and no variable in the equation. 72 is not equal to -108. This means our equation is a contradiction. There is no value that can replace our variable and make our equation true.

Skills Check:

Example #1

Identify each equation as conditional, an identity, or a contradiction. $$34 - 8x=-2(6x - 7)$$

A
Conditional
B
Identity
C

Example #2

Identify each equation as conditional, an identity, or a contradiction.

Please choose the best answer. $$4x + 2(x + 8)=6x + 16$$

A
Conditional
B
Identity
C

Example #3

Identify each equation as conditional, an identity, or a contradiction. $$-5 + 2x=\frac{1}{2}(8 + 4x)$$