Problem 16 - College Algebra

Question

Evaluate algebraic expression for the given value or values of the variable(s). \(\frac{2 x+y}{x y-2 x},\) for \(x--2\) and \(y-4\)

Step-by-Step Solution

Answer

The evaluation of the given algebraic expression for the values \(x = -2\) and \(y = 4\) is 0.

Step 1: Understand the Problem

The problem is asking to substitute the given values into the algebraic expression. The given values are \(x = -2\) and \(y = 4\). The algebraic expression is \(\frac{2 x+y}{x y-2 x}\).

Step 2: Substitute the Values

To solve this, replace \(x\) with -2 and \(y\) with 4 in the algebraic expression. Then, the expression becomes: \(\frac{2 (-2)+4}{-2*4-2*(-2)}\).

Step 3: Evaluate the Expression

Follow the order of operations (BIDMAS/BODMAS) to solve the expression. Firstly simplify inside the brackets and multiplication, then proceed with addition and subtraction. So, \(\frac{2 (-2)+4}{-2*4-2*(-2)}\) becomes \(\frac{-4+4}{-8+4}\) which in turn results in \(\frac{0}{-4}\).

Key Concepts

Substitution Method in Algebra

The substitution method in algebra is a core concept that allows us to evaluate expressions by replacing variables with their respective numeric values. This method is particularly useful when dealing with equations or expressions that involve more than one variable.

To employ the substitution method, follow these steps:

• Identify the variables in the expression.
• Determine the value of each variable from the given information.
• Replace each variable in the expression with its corresponding value.
• Simplify the expression if necessary.

For example, in the given exercise, we're asked to evaluate \(\frac{2 x+y}{x y-2 x}\), with values \(x = -2\) and \(y = 4\). By substituting \(x\) and \(y\) with -2 and 4, respectively, we can transform the expression into a simpler numerical form that can be evaluated.

Order of Operations

To correctly evaluate algebraic expressions after substituting variables, it's crucial to follow the order of operations. In mathematics, this sequence is often remembered by the acronym BIDMAS or BODMAS, which stands for Brackets, Indices/Orders (i.e., exponents and roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

This order is important because it ensures consistency in the way expressions are simplified and solved. Ignoring this sequence can lead to incorrect results. Using the order of operations:

• Simplify expressions inside the brackets first.
• Deal with exponents and roots.
• Perform multiplication and division as they appear, from left to right.
• Finally, carry out addition and subtraction as they appear, from left to right.

So when we simplify \(\frac{2 (-2)+4}{-2*4-2*(-2)}\) in our exercise, we must first multiply -2 by 2 and 4, then add and simplify within the brackets, leading us to \(\frac{0}{-4}\).

Algebraic Expression Simplification

Simplifying algebraic expressions is a process of reducing complexity, making them easier to understand or evaluate. This typically involves combining like terms, reducing fractions, and applying the order of operations properly.

The goal of simplification is to rewrite the expression in the simplest form without changing its value. Simplifying not only makes the expressions more readable but also prepares them for further operations, such as solving equations or evaluating their values.

Here are some key steps:

• Combine like terms (terms with the same variables to the same power).
• Simplify any fractions by reducing them to the lowest terms.
• Apply the distributive property to remove parentheses.
• Cancel out terms when possible to reduce the expression to its simplest form.

In our example, the expression \(\frac{2 x+y}{x y-2 x}\) simplifies to \(\frac{0}{-4}\) after substituting the values for \(x\) and \(y\), and then further simplifies to 0, because any number divided by a non-zero number is zero.