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Chapter 1: Problem 56

For Problems 41-60, simplify each numerical expression. $$ \frac{5(-9)-6(-7)}{3} $$

Short Answer

Expert verified
The simplified expression is -1.

Step by step solution

01

Distribute and Multiply

First, distribute and compute the expressions within the parentheses. Compute \( 5 \times (-9) \) and \( -6 \times (-7) \). - \( 5 \times (-9) = -45 \)- \( -6 \times (-7) = 42 \)
02

Combine the Results

Subtract the results of the multiplications from Step 1: - \(-45 + 42 = -3\)
03

Divide by 3

Take the result from Step 2 and divide it by 3: - \( \frac{-3}{3} = -1 \)

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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 a key concept that helps simplify expressions involving parentheses. It involves multiplying each term inside the parentheses by the factor outside. Imagine the distribution process as sharing a quantity equally among terms within a group. This makes complex calculations simpler by breaking them into smaller parts.
When distributing in algebra, remember the following:
  • Distribute the multiplication to each term inside the parentheses.
  • Maintain the sign in front of each number during distribution.
  • Simplify step-by-step to avoid mistakes.
For example, in the expression given, 5 is multiplied by each term within its respective parentheses:
  • i.e., multiply 5 by each term as follows: \( 5 imes (-9) \)and \(-6 \times (-7) \).
This process allows us to handle complex parts of the expression separately, making it easier to evaluate.
Multiplication of Integers
The multiplication of integers is another foundational skill in mathematics. It is essential to manage signs correctly for accurate results. When multiplying integers:
  • If both numbers have the same sign, the result is positive.
  • If the numbers have different signs, the result is negative.
For example, consider two integers:
  • \(5 \times (-9) = -45\)
  • \(-6 \times (-7) = 42\)
In the first case, positive times negative gives a negative result. In the second case, negative times negative gives a positive result.
Being careful with these rules prevents errors and ensures accurate calculations of expressions.
Division of Integers
The division of integers follows similar rules to multiplication regarding signs:
  • When you divide two integers with the same sign, the result is positive.
  • When you divide two integers with different signs, the result is negative.
In the context of our exercise, the expression \( \frac{-3}{3} \) provides a clear example:
  • Both the dividend (-3) and the divisor (3) have opposite signs. Therefore, the result is negative.
After simplifying, you obtain \(-1\).
Understanding this concept solidifies your grasp on simplifying numerical expressions, ensuring the right outcomes when handling complex problems.

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Most popular questions from this chapter

Simplify each numerical expression. Don't forget to take advantage of the properties if they can be used to simplify the computation. $$-24+18+19-30$$

Simplify each algebraic expression by combining similar terms. $$-9 y+5 y-7 y$$

Simplify each algebraic expression and then evaluate the resulting expression for the given values of the variables. \(2 x y+6+7 x y-8\) for \(x=2\) and \(y=-4\)

Simplify each algebraic expression by combining similar terms. $$13 a b+2 a-7 a-9 a b+a b-6 a$$

State whether the expressions in each problem are equivalent and explain why or why not. \(2 x-3 y+4 z\) and \(2 x-4 z+3 y\)
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