Problem 19 - Elementary Algebra

Question

Classify each of the equations as an identity, contradiction, or conditional equation. $$ y-8=-12 $$

Step-by-Step Solution

Answer

Is it an identity, a contradiction, or a conditional equation? Answer: The given equation is a conditional equation, as it is true for y = -4 but not true for other values of y.

Step 1: Understanding the types of equations

An identity is an equation that is true for all values of its variables. A contradiction is an equation that is false for all values of its variables. A conditional equation is an equation that is true for one or more, but not all values of its variables.

Step 2: Analyze the given equation

The given equation is: $$ y - 8 = -12 $$ In this equation, y is a variable. We can determine if this equation is an identity, contradiction, or conditional equation by looking for a specific value of y that makes it true or false.

Step 3: Solving for y

To determine the status of the equation, let's solve for y: $$ y - 8 + 8 = -12 + 8 \\ y = -4 $$ Now we have found an answer for y that makes the equation true: y = -4.

Step 4: Classify the equation

Since we found a value for y (-4) that makes the equation true, we can now confidently classify the given equation as a conditional equation. The equation is true for y = -4, but not true for other values of y.

Key Concepts

Identity Equation

When we talk about equations in math, an identity equation is a very special kind. It's true for any value you plug into it. Imagine a magic box that always gives a true statement no matter what you put in! For example, an equation like \( x + 1 = x + 1 \) holds true regardless of what number you choose for \(x\).

This kind of equation tells you something that is always the same, of course within the rules of math. So if you have a problem where you're asked if it's an identity, check all the potential values. If they all work, you've got yourself an identity! This is not what we saw with \( y - 8 = -12 \) because it only holds true when \( y \) equals \(-4\).

Contradiction Equation

Now let's talk about contradiction equations. They are the complete opposite of identity equations, kind of like a statement that’s always false, such as saying, “I don't exist”! More mathematically, a contradiction equation holds no truth for any real value of the variable.

Take for instance \( x + 1 = x + 2 \). No matter what value you select for \(x\), this equation will never be true. Contradictions typically tell us that something is off when any operations or errors make variables disappear leaving incorrect statements like 0 = 5.

So if you can't find any solution, it's very likely a contradiction equation. The example \( y - 8 = -12 \) is not a contradiction because we did find a value (\( y = -4 \)) that makes it true.

Solving Equations

When it comes to solving equations, it's all about finding values that make the equation a true statement. The process often involves isolating the variable, or getting the variable by itself on one side of the equation.

Let's see how this works in our given example. We had \( y - 8 = -12 \). To solve it, we performed the same operation on both sides, adding 8. This gave us \( y = -4 \). This shows us that the equation holds true when \( y = -4 \).

In broader terms:

• Start by simplifying both sides if needed.
• Use operations to get all terms with the variable on one side.
• Perform the inverse operation (like adding when there's subtraction).
• Solve for the variable.

This approach is especially useful for identifying if an equation is conditional, like in our example. Solving them accurately helps classify each equation correctly.