Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=7\) and \(y=5\). $$\frac{2 y-x+24}{2 x-y}$$

Step-by-Step Solution

Answer

The value of the expression, when \(x=7\) and \(y=5\), is 3.

Step 1: Understand and Rewrite

The given expression is \(\frac{2 y-x+24}{2 x-y}\), where \(x=7\) and \(y=5\). Therefore, substitute these values into the expression.

Step 2: Substitute Values

The substitution of the values leads to: \(\frac{2*5-7+24}{2*7-5}\).

Step 3: Simplify Numerator

First, simplify the numerator: \(2*5-7+24 = 10-7+24 = 27\).

Step 4: Simplify Denominator

Next, simplify the denominator: \(2*7-5 = 14-5 = 9\).

Step 5: Divide Numerator by Denominator

Divide the simplified numerator by the simplified denominator: \(27 \div 9 = 3\)

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra, often applied to evaluate expressions or solve equations. When working with algebraic expressions, the method requires replacing variables with their respective numerical values. For instance, if you're asked to evaluate an expression where it's given that x=7 and y=5, you will replace each occurrence of x with 7 and y with 5 in the expression.

Through this process, an abstract expression becomes a more concrete numerical expression that can then be simplified using arithmetic operations. Care should be taken to maintain the correct order of operations, commonly recalled by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that each step, from substitution to the final simplification, is executed accurately to obtain the correct value.

Simplifying Fractions

Simplifying fractions is a useful skill that helps in reducing fractions to their simplest form, making them easier to understand and work with. To simplify a fraction, you should divide the numerator (the top number) and the denominator (the bottom number) by their greatest common factor (GCF). However, when dealing with algebraic expressions, before you look to simplify, it's essential to perform all available arithmetic operations within both the numerator and denominator.

In the context of the given exercise, after substituting the variable values, you are left with a fraction. You simplify each part of the fraction by performing the arithmetic operations as indicated—first ensuring that the operations in the numerators and denominators are correctly carried out through addition and subtraction. Only after these steps would you try to find if there are common factors that could simplify the fraction further.

Algebraic Numerators and Denominators

Dealing with algebraic numerators and denominators can at first seem challenging, but sticking to algebra fundamentals can make the process more straightforward. An algebraic fraction isn't much different from a regular fraction—it still consists of a numerator and a denominator. However, in algebraic fractions, these parts can include variables along with coefficients. The key when working with them is to handle the operations carefully, ensuring that the terms are combined correctly, according to their like terms and the relevant mathematical operations.

For example, in our exercise, after substituting the values of x and y, the algebraic numerators and denominators become plain numbers. From here, you follow basic arithmetic to combine the numbers. If the resulting expression can be simplified by division, ensuring the result is in its simplest form, do so. It's important always to remember to simplify both parts separately before trying to simplify the fraction as a whole.