Problem 149 Solve. $$ 9-3(2 x+3)+6 x=0 ... [FREE SOLUTION]

Chapter 1: Problem 149

Solve. $$ 9-3(2 x+3)+6 x=0 $$

Short Answer

Expert verified

The equation holds true for all values of $x$.

Step by step solution

Distribute the -3

First, we need to distribute the -3 to both terms inside the parenthesis. The expression becomes: \[ 9 - 3 imes (2x + 3) + 6x = 0 \]Distribute the -3:\[ 9 - 6x - 9 + 6x = 0 \]

Simplify the expression

Combine like terms in the expression. The terms $-6x$ and $+6x$ cancel each other out, leaving:\[ 9 - 9 = 0 \]

Verify the equation state

After simplification, notice that the equation reduces to:\[ 0 = 0 \]This indicates that the equation is an identity and holds true for all values of $x$.

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Key Concepts

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

Distribution in Algebra

Distribution in algebra is an essential concept that allows you to eliminate parentheses by distributing a multiplier across terms inside them. For our exercise, we distributed $-3$ across $2x + 3$.

Here's how it goes:

Start with $-3 \ imes (2x + 3)$
Distribute $-3$ to each term within the parentheses:
Multiply $-3$ with $2x$, getting $-6x$.
Multiply $-3$ with $3$, getting $-9$.

Once distributed, the problem looks like this: $9 - 6x - 9 + 6x = 0$.

Notice how distribution helps in breaking down complex expressions into simpler terms, making it easier to proceed with further algebraic operations.

Combining Like Terms

After distributing, you often have to combine like terms to simplify expressions. Like terms are those with the same variable raised to the same power. In our simplified expression, $9 - 6x - 9 + 6x$, observe the like terms:

The constant terms: $9$ and $-9$
The variable terms: $-6x$ and $+6x$

Combine them carefully:

The $-6x$ and $+6x$ cancel each other out since their sum is zero.
The $9$ and $-9$ also cancel each other, simplifying to zero.

This results in $0 = 0$, showing that everything simplifies neatly.

Combining like terms is a crucial step in cleaning up algebraic expressions and paving the way to understand their structure.

Identities in Algebra

When an equation simplifies to something always true, like $0 = 0$, it's called an identity. An identity holds for any value of the variable, meaning any value of $x$ will satisfy the equation.

This happens because:

All terms cancel each other out perfectly, leaving a true statement.
No specific solution for $x$ is needed, as the initial equation is always valid.

Understanding identities helps recognize problems where no further solving is required. They indicate a fundamental truth about the relationship between terms in the equation.

Identities serve as a reminder that not all equations solve to a specific number; some describe inherent truths applicable universally.

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91影视

Solve. $$ 9-3(2 x+3)+6 x=0 $$

Short Answer

Step by step solution

Distribute the -3

Simplify the expression

Verify the equation state

Key Concepts

Distribution in Algebra

Combining Like Terms

Identities in Algebra

One App. One Place for Learning.

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