Problem 61
Question
Find the value of each of the following expressions. $$ \frac{4(8+1)-3(-2)}{-4-2} $$
Step-by-Step Solution
Verified Answer
Answer: -7
1Step 1: Solve the expressions inside the parentheses
According to the order of operations, we first need to evaluate the expressions inside the parentheses:
$$
(8 + 1) = 9
$$
So our expression becomes:
$$
\frac{4(9)-3(-2)}{-4-2}
$$
2Step 2: Perform the multiplications
Next, we will perform the multiplications:
$$
4(9) = 36 \quad\text{and}\quad -3(-2) = 6
$$
Now our expression looks like this:
$$
\frac{36+6}{-4-2}
$$
3Step 3: Perform the addition and subtraction
Lastly, we need to perform the addition and subtraction in our expression. First, let's do the addition:
$$
36 + 6 = 42
$$
Now our expression is:
$$
\frac{42}{-4-2}
$$
Next, perform the subtraction:
$$
-4 - 2 = -6
$$
So our expression is now:
$$
\frac{42}{-6}
$$
4Step 4: Divide the fraction
Finally, let's divide the fraction to get the value of our expression:
$$
\frac{42}{-6} = -7
$$
Therefore, the value of the given expression is -7.
Key Concepts
Order of OperationsArithmetic OperationsAlgebraic Fractions
Order of Operations
Understanding the order of operations is crucial for solving algebraic expressions accurately. It's a set of rules that dictate the sequence in which operations should be performed in a mathematical expression to ensure that everyone gets the correct answer. The order is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In the given exercise, the order of operations is applied by first addressing the parentheses to simplify the expression. The next step involves multiplication or division as they appear from left to right. This approach prevents potential miscalculations that could arise from performing arithmetic operations in the wrong sequence. For instance, if you were to multiply before resolving what's within the parentheses, the result would differ and be incorrect. That's why understanding and applying the correct order of operations is a fundamental skill in algebra.
In the given exercise, the order of operations is applied by first addressing the parentheses to simplify the expression. The next step involves multiplication or division as they appear from left to right. This approach prevents potential miscalculations that could arise from performing arithmetic operations in the wrong sequence. For instance, if you were to multiply before resolving what's within the parentheses, the result would differ and be incorrect. That's why understanding and applying the correct order of operations is a fundamental skill in algebra.
Arithmetic Operations
Arithmetic operations, which include addition, subtraction, multiplication, and division, are the building blocks of algebra and serve as the foundation for solving algebraic expressions. In this exercise, multiplication was used to find the values of \(4(9)\) and \(-3(-2)\), highlighting the multiplicative property of negatives which states that the product of two negative numbers is a positive number. Following multiplication, the addition of \(36 + 6\) was necessary to combine like terms. Lastly, the entire expression was divided by \(-6\), demonstrating the importance of arithmetic operations in simplifying expressions and finding a solution, in this case, \(-7\).
Correctly applying arithmetic operations is essential for the step-by-step simplification of algebraic expressions. Each operation follows specific rules, and understanding these rules is key to correctly solving an algebraic expression.
Correctly applying arithmetic operations is essential for the step-by-step simplification of algebraic expressions. Each operation follows specific rules, and understanding these rules is key to correctly solving an algebraic expression.
Algebraic Fractions
Algebraic fractions are fractions that contain variables and/or numbers in their numerators and denominators. They can seem daunting but can be simplified using basic mathematical operations and by following the order of operations. In the given exercise, the numerator and the denominator of the fraction were treated separately using arithmetic operations before performing the final division step. This process is typical for dealing with algebraic fractions and requires a careful application of all operations to reach the correct result.
To simplify algebraic fractions, factorization and common denominators are often used alongside arithmetic operations. However, in the exercise, the key was to build up to the division step by ensuring that both the numerator and the denominator were fully simplified first. The simplified fraction\(\frac{42}{-6}\) then easily yielded the answer of \(-7\). This example illustrates that algebraic fractions are just an extension of numerical fractions and can be approached with the same methodologies.
To simplify algebraic fractions, factorization and common denominators are often used alongside arithmetic operations. However, in the exercise, the key was to build up to the division step by ensuring that both the numerator and the denominator were fully simplified first. The simplified fraction\(\frac{42}{-6}\) then easily yielded the answer of \(-7\). This example illustrates that algebraic fractions are just an extension of numerical fractions and can be approached with the same methodologies.
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