Explain the difference between a conditional equation, an identity, and a contradiction.

Name: Problem 3, Chapter 2: Precalculus: Graphs and Models
Uploaded: 2020-06-19T21:54:33-08:00
Duration: 4 min 8 s
Channel: Ernest Castorena
Description: Explain the difference between a conditional equation, an identity, and a contradiction.

Explain the difference between a conditional equation, an...

Key Concepts

Conditional Equation

A conditional equation is an equation that is true only for specific values of the variable(s) involved. It does not hold universally and requires solving the equation to determine the particular conditions or values that satisfy it. For example, equations like 2x + 3 = 7 have solutions that must be found, and outside these solutions, the equation would not be true.

Identity

An identity is an equation that is true for all values of the variable(s) in its domain. It represents a universal truth in algebra where both sides of the equation simplify to the same expression regardless of the value substituted for the variable(s). For instance, expressions like (x + 3) - x = 3 are valid for every allowed value of x.

Contradiction

A contradiction is an equation that is inherently false for all values of the variable(s), meaning there are no solutions that satisfy the equation. It represents a statement that, regardless of the input, leads to an absurd or impossible result, such as an equation simplifying to a false statement like 0 = 5.

Transcript

00:02 This time they want us to explain the difference between a identity, a conditional, and a contradiction.

00:22 Okay, let's see.

00:24 Let's go with conditional equation.

00:30 So an identity would be when we have an equation, like 2x minus 1, is equal to 2x minus 8.

00:40 And i could say solve this equation for x.

00:43 But if it turns out it's true for all choices of x, so x is true for all possible, like, real numbers, then you would say it's an identity.

00:56 Like, for all values, this is equal.

00:58 In that case, this equation is equal to that.

01:01 And that is why we have the distributive law, because it's an identity.

01:06 It allows you to this translate this equation into this one over here, because they're equivalent for any value of x.

01:13 Now, for a conditional equation, that's a bit different.

01:15 If we have like 2x minus 4 is equal to like 3x let's say plus 12 then in this case there are some values in the real numbers where this wouldn't be true right if we put in 0 then we'll end up with a negative 4 here times a 2 and over here we'll end up with let's see a 0 here so which end up with a 12 and is 2 times negative 4 equal to 12 no we know that's not true because we got negative 8 on this side so for that specific value, it didn't work.

01:59 And actually, it turns out there's only one value that will make this equation work.

02:03 So let's do our algebra to solve for that.

02:06 And we'll find conditionally the only values that make it work.

02:11 2x minus 8.

02:13 I did the distribution is equal to 3x plus 12.

02:17 Now what i can do is subtract 2x from this side.

02:20 Subtract 12 here.

02:21 So i'll have negative 20 is equal to x...