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Chapter 2: Problem 34

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ -7(5 a) $$

Short Answer

Expert verified
-35\(a\)

Step by step solution

01

Multiplication of Integers

Multiply -7 and 5 to get -35. Now, the expression becomes -35\(a\).

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Key Concepts

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

Multiplication of Integers
Multiplying integers is a fundamental concept in mathematics that involves combining two whole numbers to yield a product. The process is straightforward:
  • If one or both of the numbers are negative, the product will also be negative.
  • If both integers are positive, or both are negative, the result will be positive.
To multiply integers like \(-7 \times 5\), identify the signs first:
  • \(-7\) is negative
  • 5 is positive
Then, use the rule: different signs result in a negative product. Multiply the absolute values (ignore the signs momentarily): \(7 \times 5 = 35\). Since the initial numbers had different signs, the product is negative, resulting in \(-35\).
Understanding this basic principle aids in tackling more complex mathematical problems.
Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, variables (like \(a\), \(b\), or \(x\)), and operations (addition, subtraction, multiplication, and division). These expressions allow us to generalize mathematical concepts and find solutions for variable scenarios.
In the expression '-35\(a\)', we have:
  • -35 is the coefficient, a numerical factor multiplying the variable.
  • \(a\) is the variable, representing a value that can change.
When simplifying algebraic expressions, it often involves performing operations to combine like terms or reduce expressions to a simpler form.
Although '-35\(a\)' cannot be simplified further without additional information about \(a\), the goal is to maintain a clear structure that easily communicates the expression's components.
Integer Operations
Integer operations encompass addition, subtraction, multiplication, and division. These operations are the building blocks for more advanced concepts in mathematics.
  • **Addition/Subtraction**: Combine or subtract integers by considering their signs; same signs add, different signs subtract.
  • **Multiplication/Division**: As covered earlier, multiply or divide considering the rules for signs to determine whether the outcome is positive or negative.
It is essential to practice these operations to ensure competence in handling both straightforward and complex mathematical problems.
For example, working through \(-7 \times 5\) develops the skill to manage integer operations involving different signs.
These operations are crucial not only for solving algebraic expressions but for understanding the relationships between numbers in various mathematical contexts.

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Most popular questions from this chapter

In Exercises 63-68, translate the verbal phrase into an algebraic expression. Simplify the expression. $$ \text { The square of } x \text { added to the product of } x \text { and } x+1 $$

In Exercises 33-38, justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} \frac{2}{3} x &=12 \\ \frac{3}{2}\left(\frac{2}{3} x\right) &=\frac{3}{2}(12) \\ x &=18 \end{aligned} $$

In Exercises 17-26, determine whether an algebraic expression or an algebraic equation is given. $$ x-4 $$

In Exercises 27-32, solve the equation. $$ x-8=5 $$

A train travels at an average speed of 50 miles per hour. How long will it take the train to travel 350 miles?
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