Problem 13 Perform the indicated operations... [FREE SOLUTION]

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Precalculus: Mathematical for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 1: Problem 13

Perform the indicated operations and simplify. $$8(2 x+5)-7(x-9)$$

Short Answer

Expert verified

The simplified expression is $9x + 103$.

Step by step solution

Distribute the 8 into the first parentheses

Multiply 8 by each term inside the parentheses: $8(2x + 5)$.\[8 \times 2x = 16x\]\[8 \times 5 = 40\]This results in: $16x + 40$.

Distribute the -7 into the second parentheses

Multiply -7 by each term inside the parentheses: $-7(x - 9)$.\[-7 \times x = -7x\]\[-7 \times -9 = 63\]This results in: $-7x + 63$.

Combine like terms

Now combine all the terms from Step 1 and Step 2:$16x + 40 - 7x + 63$.Combine the $x$ terms: $16x - 7x = 9x$.Combine the constant terms: $40 + 63 = 103$.

Write the final simplified expression

The expression simplifies to:\[9x + 103\].

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Key Concepts

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

Distributive Property

When faced with expressions like $8(2x + 5)$ or $-7(x - 9)$, the distributive property is your best friend. This mathematical rule helps you multiply a single number by each term within parentheses.
To apply it, you multiply the outer number by each term inside. For example:

Distribute $8$ to $2x$ and $5$. This gives $8 \times 2x = 16x$ and $8 \times 5 = 40$.
Similarly, distribute $-7$ to $x$ and $-9$. This gives $-7 \times x = -7x$ and $-7 \times -9 = 63$.

Remember, when you distribute, you're ensuring every term inside the parenthesis is touched by the outside number.
Attention to signs is crucial here, as multiplication with positive and negative numbers affects the outcome greatly!

Combining Like Terms

After using the distributive property, you'll often find different terms that can be combined. These are known as 'like terms'.
Like terms are those terms that contain exactly the same variables raised to the same power, with or without coefficients. For instance, after distribution, our expression had terms like $16x$ and $-7x$.
Combining like terms involves:

Grouping terms with the same variable component: $16x$ and $-7x$ can be combined because they both contain the variable $x$.
Adding or subtracting the coefficients of these grouped terms: $16x - 7x = 9x$.
Grouping constant numbers together: $40$ and $63$ sum up to $103$.

By combining like terms, you simplify the expression, making it easier to further manipulate or solve.

Simplification of Expressions

The final step in handling algebraic expressions like $8(2x+5)-7(x-9)$ is simplifying them. This involves reducing the expression to its simplest form by applying all the steps we鈥檝e discussed.
Once you've distributed and combined like terms, draw everything together to present the final, simplified expression:

Start with all the terms: $16x + 40 - 7x + 63$.
Combine the like terms: this gives $9x$ for the 'x' terms and $103$ for the constants.

So, this expression ultimately simplifies to $9x + 103$.
This simplification process is crucial as it reduces complexity and provides clarity, making the expression straightforward and user-friendly.

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91影视

Perform the indicated operations and simplify. $$8(2 x+5)-7(x-9)$$

Short Answer

Step by step solution

Distribute the 8 into the first parentheses

Distribute the -7 into the second parentheses

Combine like terms

Write the final simplified expression

Key Concepts

Distributive Property

Combining Like Terms

Simplification of Expressions

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