Problem 1 Simplify the expression. $$ ... [FREE SOLUTION]

Chapter 0: Problem 1

Simplify the expression. $$ -3(8-4 x)-\frac{1}{3}(15 x-6 z) $$

Short Answer

Expert verified

-24 + 7x + 2z

Step by step solution

Distribute the Constants

Start by distributing the constants $-3$ and $\frac{1}{3}$ to each term inside their respective parentheses. \[-3(8-4x) \rightarrow -3 \times 8 + (-3) \times (-4x) = -24 + 12x\] \[\frac{1}{3}(15x-6z) \rightarrow \frac{1}{3} \times 15x + \frac{1}{3} \times (-6z) = 5x - 2z\]

Write the Expression Containing the Distributed Values

Combine the distributed values into one resulting expression. \[-24 + 12x - 5x + 2z\]

Combine Like Terms

Combine the like terms (which are the terms with $x$ and the constant terms). \[12x - 5x = 7x\] Therefore, the expression becomes \[-24 + 7x + 2z\]

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Key Concepts

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

Distributive Property

Understanding the distributive property is the first step in simplifying many algebraic expressions. This property tells us how to multiply a single term across terms inside parentheses. When you see an expression like $-a(b+c)$, you distribute the $-a$ to both $b$ and $c$. For example:

$-3(8 - 4x) = -3 \times 8 + (-3) \times (-4x)$
This results in $-24 + 12x$.

Similarly, if you have a fraction like $\frac{1}{3}(15x - 6z)$, you distribute $\frac{1}{3}$ to both terms:
$\frac{1}{3} \times 15x + \frac{1}{3} \times (-6z) = 5x - 2z$.

Remember, always distribute outside terms to every term inside the parentheses.

Combining Like Terms

Once you have used the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable part. Here is an example from the earlier step:

\[-24 + 12x - 5x + 2z\]
This expression has like terms with the variable $x$:
\[12x \text{ and } -5x\].

To combine them, just add or subtract their coefficients:
\[12x - 5x = 7x\].

Now, the expression looks like this:
\[-24 + 7x + 2z\].
You'll notice terms like $-24$ and $2z$ can't be combined with others, because they don't share the same variables.

Algebraic Simplification

Finally, algebraic simplification brings everything together. Simplifying an expression means to make it as simple as possible. For the expression \-24 + 7x + 2z\:

* We've already distributed and combined like terms.
* Now, ensure it is in its simplest form.

There are no more like terms to combine in $-24 + 7x + 2z$.
Therefore, this expression is fully simplified. This process helps in solving equations efficiently and makes the expression more manageable.

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91影视

Simplify the expression. $$ -3(8-4 x)-\frac{1}{3}(15 x-6 z) $$

Short Answer

Step by step solution

Distribute the Constants

Write the Expression Containing the Distributed Values

Combine Like Terms

Key Concepts

Distributive Property

Combining Like Terms

Algebraic Simplification

One App. One Place for Learning.

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