Problem 48 - College Algebra

Question

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

Step-by-Step Solution

Answer

The evaluated expression for x=-2 and y=4 is 0.

Step 1: Substitution

Substitute the given values of x as -2 and y as 4 into the algebraic expression: \(\frac{2(-2)+4}{(-2)(4)-2(-2)}\).

Step 2: Simplify Numerator

Next, simplify the expression in the numerator. You have \(2(-2) + 4\) which is \(-4 + 4 = 0\).

Step 3: Simplify Denominator

Next, simplify the expression in the denominator. You have \(-2(4) - 2(-2)\) which is \(-8 + 4 = -4\).

Step 4: Simplify Entire expression

Now apply the computed numerator and denominator to simplify the entire expression: \(\frac{0}{-4}\). The result is 0. Remember that zero divided by any number is zero.

Key Concepts

Substitution Method

Understanding the substitution method is crucial for solving algebraic expressions involving variables. This technique allows you to plug specific values into variables and simplify the expression using arithmetic operations. In our problem, the expression \(\frac{2x+y}{xy-2x}\) is evaluated by replacing \(x\) with \(-2\) and \(y\) with \(4\). Here's how to approach substitution:

• Start by rewriting the original expression with parentheses, inserting the given values for each variable: \(\frac{2(-2)+4}{(-2)(4)-2(-2)}\). This helps avoid mistakes with signs and operations.
• Ensure that each term of the expression is calculated individually. By substituting, you transform an algebraic expression into an arithmetic one.

The substitution method helps simplify the process by reducing variable terms to numerical values for easier calculations. This method is universal and can be used for various expressions with differing numbers of variables.

Simplifying Expressions

The key to simplifying expressions lies in handling each component carefully and methodically. After substitution, algebraic expressions turn into arithmetic ones, which need to be simplified. Let's explore:

• Simplify the Numerator: After substitution, the numerator \(2(-2) + 4\) becomes \(-4 + 4\). Simplifying it gives you \(0\).
• Simplify the Denominator: Similarly, the denominator \((-2)(4) - 2(-2)\) simplifies to \(-8 + 4\), resulting in \(-4\).

When simplifying, always focus one section at a time—numerator and then denominator. Performing operations such as addition, subtraction, multiplication, and division accurately is important for reaching the correct result.
Simplifying expressions leads to an easier understanding of the entire problem and ensures accuracy in final outcomes.

Numerator and Denominator Simplification

Simplifying the fraction's numerator and denominator separately helps manage complex expressions effectively. Each part of the fraction should be simplified before attempting to find the fraction's value.

• The Numerator in our example, \(2x + y\), when simplified after substitution becomes \(0\).
• The Denominator, \(xy - 2x\), after substitution and simplification, results in \(-4\).

Once both numerator and denominator are simplified, the final fraction \(\frac{0}{-4}\) represents a straightforward solution, which is \(0\).
This methodical separation of simplifying the numerator and denominator prevents the common mistake of wrong calculations. Remember, a numerator of zero indicates that the fraction equals zero, regardless of what the denominator is (provided it's not zero). This concept is fundamental in algebra to determine the value of expressions efficiently and correctly.