Problem 11 - Elementary Algebra

Question

Find the value of \(\frac{-7(4-8)+2(1-11)}{-5(1-6)-17}\).

Step-by-Step Solution

Answer

Question: Evaluate the following expression: \(\frac{-7(4-8)+2(1-11)}{-5(1-6)-17}\) Answer: The value of the given expression is \(1\).

Step 1: Evaluate expressions inside parentheses

First, we need to calculate the values inside the parentheses. For the numerator, we have \((4-8)\) and \((1-11)\). For the denominator, we have (\((1-6)\)). \(\frac{-7(4-8)+2(1-11)}{-5(1-6)-17} = \frac{-7(-4)+2(-10)}{-5(-5)-17}\)

Step 2: Multiply the numbers

Now, let's multiply the numbers: \(-7 \times -4 = 28\), \(2 \times -10 = -20\), \(-5 \times -5 = 25\). So, the expression becomes: \(\frac{28-20}{25-17}\)

Step 3: Simplify the expression

Now, let's simplify the expression by performing the addition and subtraction operations: \(\frac{28-20}{25-17} = \frac{8}{8}\)

Step 4: Divide the numbers

As a final step, let's divide the numerator by the denominator: \(\frac{8}{8} = 1\) Thus, the value of the given expression is \(1\).

Key Concepts

Order of Operations

Understanding the order of operations is critical when evaluating algebraic expressions. This sequence determines the procedure to solve math problems accurately. The standard order is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), often abbreviated as PEMDAS.

Let's apply this to an example expression: \(\frac{-7(4-8)+2(1-11)}{-5(1-6)-17}\). According to PEMDAS, we first tackle the parentheses \(4-8\), \(1-11\), and \(1-6\), then move on to multiplication \(\times\) and division \(\div\), and lastly perform addition \(+\) and subtraction \(–\). Following these steps ensures that expressions like this one are simplified correctly, leading to the accurate result of \(1\).

Remembering to follow the order of operations can prevent errors and confusion when handling complex expressions.

Simplifying Expressions

To simplify an algebraic expression means to reduce it to its simplest form. This involves a multitude of steps such as combining like terms, using the distributive property, and canceling out terms where possible.

For example, in the expression \(\frac{28-20}{25-17}\), we combine the like terms by subtracting \(20\) from \(28\) and \(17\) from \(25\). This results in the much simpler expression \(\frac{8}{8}\), which can then be simplified further to \(1\). When simplifying, it's important to always double-check that no further simplification can be done. Keeping expressions as simple as possible can make it much easier to manipulate them in more complex equations.

Arithmetic Operations in Algebra

When engaging in arithmetic operations in algebra, we extend the simple operations of addition, subtraction, multiplication, and division to include variables along with numbers. However, it is fundamental to acknowledge that the underlying principles of these operations remain unchanged.

In the context of our example, \(\frac{-7(4-8)+2(1-11)}{-5(1-6)-17}\), we perform multiplication \(\-7 \times -4\) and \(2 \times -10\), and division when we simplify \(\frac{8}{8}\) to \(1\). In algebra, we must consider the order of operations and the proper handling of negative and positive signs. Especially with negative signs, mistakes can easily occur if they are not systematically addressed, such as understanding that multiplying two negative numbers results in a positive number.

By mastering arithmetic operations in algebra, students can tackle a broader spectrum of calculations and find solving algebraic expressions much more manageable.