Problem 47 - College Algebra

Question

evaluate each algebraic expression for the given value of the variable or variables. $$ \frac{2 x+3 y}{x+1} ; x=-2 \text { and } y=4 $$

Step-by-Step Solution

Answer

The evaluated value of the algebraic expression for \(x = -2\) and \(y = 4\) is \(-8\).

Step 1: Replace variables with given values

This requires substitution of the given values into the equation. So, \(x\) is replaced with \(-2\) and \(y\) is replaced with \(4\) in our expression, we get: \(\frac{2(-2)+3(4)}{(-2)+1}\).

Step 2: Perform Multiplication

This is where BODMAS comes into play. 'Multiplication' comes before 'Addition' in the order of operations. So we'll do that first inside the numerator and denominator to get: \(\frac{-4+12}{-2+1}\).

Step 3: Execute Addition/Subtraction

The next step in BODMAS is 'Addition/Subtract'. Both the numerator and denominator need simplification separately which gives: \(\frac{8}{-1}\).

Step 4: Perform Division

Finally, we will perform the 'Division' in the fraction to arrive at the numerical result. Dividing 8 by -1 gives: \(-8\).

Key Concepts

Order of Operations

When evaluating algebraic expressions, the order of operations is crucial for obtaining the correct result. The order of operations can be remembered with the acronym BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). It's a set of rules that indicates the order in which different operations should be done to ensure a consistent result.

• Begin by solving anything inside brackets or parentheses.
• Next, handle exponents or "orders."
• Proceed with division and multiplication from left to right.
• Finish with addition and subtraction, also from left to right.

In our exercise, after substituting the values of variables, we follow the BODMAS rule. First, perform the multiplication within the numerator and denominator: \(2(-2)\) and \(3(4)\) become \(-4\) and \(12\) respectively. Then, move on to simplifying by subtraction and addition, left to right. This ensures clarity and prevents mistakes. Only by following these steps meticulously, can we arrive at the correct solution of \(-8\).

Substitution Method

The substitution method is straightforward and important when working with algebraic expressions. It involves replacing variables with the given numerical values. This allows the expression to be simplified down to a numeric calculation rather than dealing with abstract variables. Let's break down how substitution takes place:

• Identify each variable in the expression.
• Replace each variable with the numerical value provided.
• Simplify the resulting expression following the order of operations.

In the problem provided, we have the expression \(\frac{2x+3y}{x+1}\). Given that \(x = -2\) and \(y = 4\), the substitution step involves replacing \(x\) with \(-2\) and \(y\) with \(4\). This transforms the expression into \(\frac{2(-2)+3(4)}{(-2)+1}\). With the correct substitution, you prepare the expression for further simplification and ultimately solve it.

Algebraic Simplification

Algebraic simplification is the process of making an expression more manageable and easier to understand. After substitution, the expression often needs simplification to reach the final answer. This might involve combining like terms, performing arithmetic operations, and eliminating any complexity that doesn't affect the overall value of the expression.Here's a step-by-step process to simplify:

• After substitution, perform all internal multiplications in the expression.
• Combine terms or numbers, specifically focusing on like terms in the numerator and denominator.
• Carry out the final operation—often this is addition or subtraction.

In our example, we started with \(\frac{2(-2)+3(4)}{(-2)+1}\), which simplified through successful execution of the multiplication gives \(\frac{-4+12}{-2+1}\). The fraction further simplifies to \(\frac{8}{-1}\) by adding or subtracting in both the numerator and the denominator. Finally, the division of \(8\) by \(-1\) results in \(-8\). Such steps are essential not just to arrive at a numerical answer but to grasp how algebra works in breaking down expressions.