Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=7\) and \(y=5\). $$\frac{50}{y}-\frac{14}{x}$$

Step-by-Step Solution

Answer

The evaluated value of the given expression is \(8\)

Step 1: Substitute the values

First, substitute the given values for \(x\) and \(y\) into the expression. This gives us:\[\frac{50}{5} - \frac{14}{7}\]

Step 2: Simplify each fraction

Next, simplify each fraction separately. This gives us:\[10 - 2\]

Step 3: Subtract the two numbers

Finally, subtract the two numbers to get the result. This gives us:\[8\]

Key Concepts

Expressions

Expressions in algebra are mathematical phrases that can include numbers, variables, and operators. Variables like \(x\) and \(y\) are placeholders that can represent different values. The core idea is to evaluate these expressions by using given values for these variables. In our exercise, the expression is \(\frac{50}{y} - \frac{14}{x}\). Here, \(x=7\) and \(y=5\) are values assigned to the variables. By replacing the variables in the expression with these respective numbers, we can simplify the expression into a numerical form and solve it easily.

Substitution

Substitution is a method used in algebra to replace variable placeholders with actual numbers. When solving algebraic problems, this step is crucial to make the expression solvable. For example, in our exercise, the expression \(\frac{50}{y} - \frac{14}{x}\) becomes \(\frac{50}{5} - \frac{14}{7}\) after substitution.
This step essentially transforms a general expression into a specific numerical calculation. After this, the expression no longer contains any variables and can be simplified through arithmetic operations.

Simplification

Simplification is the process of breaking down expressions into simpler or more manageable forms. In the context of our problem, once the values of \(x\) and \(y\) have been substituted into the expression, each fraction can be simplified separately.

• The part \(\frac{50}{5}\) simplifies to 10 because 50 divided by 5 is 10.
• The part \(\frac{14}{7}\) simplifies to 2 because 14 divided by 7 is 2.

After these simplifications, the expression \(\frac{50}{5} - \frac{14}{7}\) reduces to \(10 - 2\). The simplification makes the expression easier and more straightforward to solve.

Fraction Operations

Fractions involve a numerator and a denominator, and fraction operations are all about performing arithmetic operations such as addition, subtraction, multiplication, and division on these fractions. In our exercise, we deal with the subtraction of two fractions post-simplification.

• First, each fraction is simplified: \(\frac{50}{5}\) as 10, and \(\frac{14}{7}\) as 2.
• Then, the operation of subtraction is performed: 10 - 2.

Understanding how to effectively manage fraction operations is crucial. It allows one to handle more complex expressions with ease by breaking them down into simple arithmetic operations like those completed in this exercise.