Problem 74 - Algebra 1

Question

Evaluate the expression. $$\frac{3}{4} \cdot[-7 \cdot(-4-6)+30]-11$$

Step-by-Step Solution

Answer

The result of the expression is 64

Step 1: Simplify Inside the Bracket

First, simplify the expression inside the bracket. The expression -4 - 6 equals to -10.

Step 2: Multiplication

Multiply the result obtained from Step 1 with -7. So, -7 * -10 equals to 70.

Step 3: Addition

Add the result we got from the last step with 30, 70 + 30 equals to 100.

Step 4: Multiply the Fraction

Multiply the result from step 3 with 3/4. We get, 3/4 * 100 equals to 75.

Step 5: Subtraction

After we get the result, we subtract 11 from it. So, 75 - 11 equals to 64.

Key Concepts

Order of Operations

When solving any mathematical expression, it's important to follow the order of operations to ensure that the solution is accurate. The acronym PEMDAS can help you remember the correct order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). For example, if you have an expression like \(3 + 4 \times 2\), you would first perform the multiplication \(4 \times 2 = 8\), and then the addition \(3 + 8 = 11\).
In our original exercise, you start by simplifying the expression within the brackets because parentheses (or in this case, a full bracket) are the first priority. After that, any multiplication must be completed before moving on to addition or subtraction, ensuring we tackle operations in the correct sequence to avoid errors.

Multiplication in Algebra

Multiplication in algebra often involves a combination of numbers and variables. In this context, you need to remember that multiplying negative numbers results in a positive product. For example, \(-7\) times \(-10\) equals \(70\) since two negatives make a positive.
Moreover, when dealing with fractions like \(\frac{3}{4} \cdot 100\), it's helpful to think of it as finding a fraction of a number, which involves multiplying the numerator by the number and then dividing by the denominator. This gives us \(75\).
Using clear steps for multiplication in algebra helps to keep calculations straightforward and manageable, reducing errors and confusion.

Simplifying Expressions

Simplifying expressions means making them as straightforward as possible while retaining their value. This involves combining like terms, performing operations in brackets, and reducing fractions or simplifying coefficients. By simplifying, you make computation easier and prepare the expression for further operations.
In our original exercise, simplifying took place in several stages: first, inside the bracket \((-4 - 6 = -10)\), then simplifying further by dealing with the multiplication of \(-7 \times -10\), and finally when you multiply the result by \(\frac{3}{4}\). This progressive simplification is crucial for accurately evaluating expressions.

Evaluating Expressions

Evaluating an expression means finding its numerical value. This is the end goal where you've applied the correct order of operations, simplifying and multiplying as needed. Evaluation gives you a final, simplified outcome that represents the original expression's value.
After carrying out all necessary operations in the original exercise, we end up with \(75\), from which we then subtract \(11\) as the final operation, yielding \(64\).
Evaluating ensures you have the right answer by the end of your mathematical journey. Each step builds logically upon the previous one, ensuring clarity and correctness throughout the process.