Problem 23 - Introductory Algebra for College Students

Question

Evaluate each expression for \(x=7\) and \(y=5\). $$\frac{2 x-y+6}{2 y-x}$$

Step-by-Step Solution

Answer

The value of the expression when \(x = 7\) and \(y = 5\) is 5.

Step 1: Substitute the values of x and y

Replace \(x\) with 7 and \(y\) with 5 in the given expression. This gives \[\frac{2(7) - 5 + 6}{2(5) - 7}\]

Step 2: Simplify the Numerator

Now, follow the order of operations to solve the expression in the numerator. This gives \[\frac{14 - 5 + 6}{2(5) - 7} = \frac{15}{2(5) - 7}\]

Step 3: Simplify the Denominator

Repeat the order of operations to solve the expression in the denominator. This gives \[\frac{15}{10 - 7} = \frac{15}{3}\]

Step 4: Simplify the Division

Now, complete the division to find the answer. This gives \(5\).

Key Concepts

Order of Operations

When simplifying an algebraic expression, it is crucial to follow the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule allows for consistency in solving mathematical equations and ensures that everyone arrives at the same answer.

For instance, in the expression \(\frac{2x - y + 6}{2y - x}\), you must firstly address the operations inside the parentheses. This means performing multiplication or division before moving on to addition or subtraction. It's common for students to overlook this step, but understanding and applying the correct order of operations is key to arriving at the correct simplification of the expression.

Substitute Variables

Substituting variables is a fundamental skill that allows you to evaluate expressions by replacing letters (variables) with actual numbers. To accurately substitute variables, you must first understand what each variable represents and then replace it with its corresponding value.

In our exercise, the variables \(x\) and \(y\) are given the values 7 and 5, respectively. This substitution clarifies the expression, transforming it from an abstract equation to a solvable numerical problem. Ensure that you replace each instance of the variable with the specific value assigned to it, as overlooking this can lead to incorrect results. After substitution, you're ready to perform the necessary arithmetic operations to simplify the expression further.

Solving Rational Expressions

Rational expressions involve ratios of polynomial expressions. Solving them often requires a careful application of both substitution of variables and order of operations, as demonstrated in the step by step solution we've discussed.

When simplifying rational expressions, it's essential to perform operations for the numerator and the denominator separately until the point of division. In the given example, \(\frac{15}{3}\), the final step is to divide the numerator by the denominator, which yields 5. Understanding the order of operations and accurate variable substitution are both pivotal aspects of successfully simplifying rational expressions. As a result, mastering these skills will allow you to work through algebraic problems more efficiently and with increased confidence.