Identifying Types of Equations Lesson

As we learned in Algebra 1, we can categorize our equations as: conditional, identity, or contradiction.

Conditional Equations

Most equations we work with are conditional equations. A conditional equation is true only under certain conditions. Let's take a look at an example.
Example 1: Determine if the equation is conditional, an identity, or a contradiction.
-8x - 38 = -5(1 - 5x)
Let's begin by removing parentheses on the right side:
-8x - 38 = -5 + 25x
Now we can move all the variable terms to the left side and all the constants to the right side:
-8x - 25x = -5 + 38
-33x = 33
Divide each side of the equation by -33, this will isolate x:
(-33/-33)x = (33/-33)
x = -1
Since we have only one solution, we can say this equation is a conditional equation. It is true when (-1) replaces x, but false for any other number.

Contradiction Equation

From time to time, we will run across an equation with no solution. This equation type is known as a contradiction. Let's take a look at an example.
Example 2: Determine if the equation is conditional, an identity, or a contradiction.
-4x - 13 = -(8 + 4x)
Let's begin by removing parentheses on the right side:
-4x - 13 = -8 - 4x
Now we can move all the variable terms to the left side and all the constants to the right side:
-4x + 4x = -8 + 13
0 = 5 (false)
When we end up with a false statement, we can stop and say our equation has "no solution". Since we studied sets earlier in our course, we can also use the empty set symbol to show our solution set is empty, meaning it contains no elements:
∅

Identity Equation

Lastly, we will see equations that have an infinite number of solutions. Let's take a look at an example.
Example 3: Determine if the equation is conditional, an identity, or a contradiction.
2x + 5(x - 8) = -40 + 7x
Let's begin by removing parentheses on the left side:
2x + 5x - 40 = -40 + 7x
Now we can simplify the left side:
7x - 40 = -40 + 7x
Before we go any further, we should notice that the two sides have exactly the same terms (7x and -40). This means whatever value is plugged in for x will always yield a true statement.
If we continue with our normal procedure, we would next move all the variable terms to the left and all the constants to the right:
7x - 7x = -40 + 40
0 = 0 (true)
When we end up with a true statement and no variable, we can stop and say our equation has "an infinite number of solutions".

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