Problem 18 Identify each equation as linear... [FREE SOLUTION]

Chapter 11: Problem 18

Identify each equation as linear or quadratic. $$10-2(3 d+1)=5 d+19$$

Short Answer

Expert verified

The simplified equation is $8 - 6d = 5d + 19$. This equation is linear since it only contains the first-degree variable d without any square terms.

Step by step solution

Distribute -2 to the terms inside the parenthesis

In this step, we will distribute -2 to both terms inside the parenthesis (3 d + 1): \[-2(3d) + (-2)(1) = -6d - 2\] Now, the equation becomes: \[10 - 6d - 2 = 5d + 19\]

Combine like terms on the left side of the equation

We can combine the constant terms (10 and -2) on the left side of the equation: \[8 - 6d = 5d + 19\]

Identify the equation as linear or quadratic

Now that we have simplified the equation, we can see its form: \[8 - 6d = 5d + 19\] This equation contains only a first-degree variable (d) without any square terms, so it is a linear equation.

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Key Concepts

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

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It allows you to multiply a single term across two or more terms inside parentheses. This step is crucial in making equations easier to solve.
In the problem given, the distributive property is applied by multiplying $-2$ with each term inside the parentheses $(3d + 1)$.
This yields:

$-2 \times 3d = -6d$
$-2 \times 1 = -2$

Thus, the expression $-2(3d + 1)$ expands to $-6d - 2$. Applying the distributive property is a powerful technique to transform complex expressions into simpler forms, making them easier to handle in subsequent steps.

Combining Like Terms

After distributing, the next step is to look for terms you can combine, which is called combining like terms. Like terms are terms that have the same variable raised to the same power.
In our example, once we distribute, the equation becomes $10 - 6d - 2 = 5d + 19$.
On the left side of the equation, you can combine the constant numbers $10$ and $-2$:

This results in the simplified expression: $8 - 6d$.

By combining like terms, you simplify the equation further, reducing unnecessary complexity and making it easier to solve or analyze. This step ensures all terms are logically grouped, clearing the way for solving the equation.

First-Degree Polynomial

A first-degree polynomial is an expression that involves variables raised to the power of one. These are often referred to as linear equations.
Linear equations take the general form:

$ax + b = cx + d$ where $a$, $b$, $c$, and $d$ are constants.

In this problem, the equation simplified to $8 - 6d = 5d + 19$, where $d$ is the variable.
This is a first-degree polynomial because the highest power of the variable $d$ is one. There are no squared terms or higher powers, confirming that the equation is a linear equation.
Understanding that an equation is linear helps in predicting behavior and solutions, as linear equations graph as straight lines and often have simple solutions.

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91影视

Identify each equation as linear or quadratic. $$10-2(3 d+1)=5 d+19$$

Short Answer

Step by step solution

Distribute -2 to the terms inside the parenthesis

Combine like terms on the left side of the equation

Identify the equation as linear or quadratic

Key Concepts

Distributive Property

Combining Like Terms

First-Degree Polynomial

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