Problem 40 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Solve the equation symbolically. (b) Classify the equation as a contradiction, an identity, or a conditional equation. $$ 7-9 z=2(3-4 z)-z $$

Step-by-Step Solution

Answer

The solution to the equation is a contradiction; the equation simplifies to 7 = 6, which is never true.

Step 1: Simplify Both Sides

First, let's simplify both sides of the equation. Begin by expanding the right-hand side of the equation: \[ 7 - 9z = 2(3 - 4z) - z \] Distribute the 2 across the terms inside the parentheses: \[ = 6 - 8z - z \] Simplifying the right-hand side, we combine like terms: \[ = 6 - 9z \] Now the equation is: \[ 7 - 9z = 6 - 9z \]

Step 2: Isolate Variable Terms

Next, isolate terms with the variable on one side. We notice that \( -9z \) is present on both sides. So, if we add \( 9z \) to both sides, we get:\[ 7 - 9z + 9z = 6 - 9z + 9z \] This simplifies to: \[ 7 = 6 \]

Step 3: Analyze the Result

The equation simplifies to \( 7 = 6 \), which is not true. This equation is always false, regardless of the value of the variable \( z \). Therefore, the equation is a contradiction as the statement is never true.

Key Concepts

Understanding Contradictions in Equations

When working with equations, sometimes you'll encounter something called a contradiction. This is when, after simplifying the equation, you end up with a statement that is always false. For instance, in our exercise, we ended with the statement \( 7 = 6 \). Such an equation does not depend on any variable, and there isn't a value you can substitute to make it true. It means there are no solutions to the equation.

- A contradiction arises when two sides of an equation cannot be made equal.- Even after removing all variables, the statement remains false.- You can recognize a contradiction when final simplification leads to a false statement like \( 3 = 5 \) or \( 0 = 10 \).

Understanding contradictions helps you quickly know when an equation doesn't have solutions, saving time that might be spent seeking a non-existent answer.

What is an Identity in Equations?

An identity in equations is a special type of equation that, when simplified, holds true no matter what value is substituted for the variable. Think of it as a universal truth for that equation. For example, the equation \( 3(x + 2) = 3x + 6 \) is an identity because, no matter what number you use for \( x \), both sides will always be equal.

- An identity gives infinite solutions, as any value plugged in for the variable works.- These equations are often used to express mathematical truths and properties.- Recognizing an identity can be helpful, as it confirms you’ve verified a fundamental truth, like distributing or simplifying expressions correctly.

By identifying identities, you grasp that certain equations aren't about solving for one number, but rather showing a consistent relationship between expressions.

Conditional Equations and Their Characteristics

Conditional equations are a bit different from contradictions or identities. These equations are true under certain conditions but not true universally. You might end up with an equation where, once simplified, it equals a true statement only if the variable takes particular values. For example, the equation \( x + 2 = 5 \) is only true when \( x = 3 \).

- Conditional equations yield specific solutions or numbers that satisfy the equation.- Unlike identities, they do not have infinite solutions and aren’t false like contradictions.- Finding solutions involves isolating the variable and solving for the exact numbers that work.

Knowing about conditional equations helps you understand that many equations relate to particular situations or conditions and the solutions derived apply specifically to those scenarios.