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Chapter 5: Problem 14

Classify each equation as an identity or a contradiction. $$ 3(2 x-4)-2(3 x+1)+14=0 $$

Short Answer

Expert verified
Answer: The given equation is an identity.

Step by step solution

01

Simplify the equation

First, we simplify the given equation by distributing the numbers inside the parentheses and then combining like terms. $$ 3(2 x-4)-2(3 x+1)+14 = 0 $$
02

Distributing the numbers inside the parentheses

Next, we distribute the numbers inside the parentheses. $$ (3*2x - 3*4) - (2*3x + 2*1) + 14 = 0 $$ $$ (6x - 12) - (6x + 2) + 14 = 0 $$
03

Combining like terms

Now, we combine the like terms to simplify the equation further. $$ (6x - 6x) + (-12 - 2 + 14) = 0 $$ $$ 0x + 0 = 0 $$
04

Determine the equation type

Since the simplified equation is always true (0 = 0) regardless of the value of x, the given equation is an identity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Equations
Simplifying equations is an essential skill in algebra that helps to solve them more easily. To simplify an equation, you need to make it shorter and more manageable, often by reducing expressions to their simplest terms.

Here are some steps you might take:
  • Clear parentheses by applying the distributive property to eliminate them.
  • Combine like terms, which means to group and simplify the terms that have the same variable part.
  • Eliminate fractions by multiplying every term by the least common denominator.
The goal is to rewrite the equation in a much simpler form until it is ready to be solved. In our example, 3(2x-4)-2(3x+1)+14=0, we would apply these steps to reach a point where we could easily determine if the equation is an identity or a contradiction.
Distributive Property
The distributive property is a fundamental principle that comes in handy when simplifying equations. It's a way of multiplying a single term and two or more terms inside a set of parentheses. It states that:\[a(b + c) = ab + ac\]

In our original equation, 3(2x-4)-2(3x+1)+14=0, the distributive property is used twice:
  • First, apply to \(3(2x-4)\) resulting in \(6x - 12\).
  • Second, apply to \(-2(3x+1)\) resulting in \(-6x - 2\).
By distributing the numbers inside the parentheses, we break down the equation into simpler parts, ensuring that every term is properly multiplied. This makes the next step of combining like terms straightforward.
Combining Like Terms
Combining like terms is a critical step in simplifying expressions and equations. Like terms are terms whose variables and their exponents are the same. By combining them, we group and simplify parts of the expression that can be added or subtracted together.

In our example equation:
  • The terms \(6x\) and \(-6x\) are like terms. When combined, they equal zero: \(6x - 6x = 0x\).
  • The constant terms like \(-12\), \(-2\), and \(+14\) are also combined to get a total constant: \(-12 - 2 + 14 = 0\).
Through combining like terms, we achieved \(0x + 0 = 0\). This shows clearly whether the equation is always true or not, aiding in classifying the equation as an identity or a contradiction.

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Most popular questions from this chapter

Write "identity," "contradiction," or "conditional." If you can, find the solution by making an educated guess based on your knowledge of arithmetic. $$ 5 a=25 $$

For the following problems, solve the linear equations in two variables. $$ -x-y=0, \text { if } y=\frac{14}{3} $$

Solve the inequalities. $$ \frac{b}{-3} \leq 4 $$

Solve \(3 y-6 x=12\) for \(y\)

Find the product. \((4 x-1)(3 x+5)\).
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