Problem 21 - Algebra for College Students

Question

Find the value of the indicated variable. Round approximate answers to three decimal places. Find \(f\) if \(M=10, F=5,\) and \(M=\frac{F}{f}\)

Step-by-Step Solution

Answer

f = 0.5

Step 1: Write Down the Given Equation

The given equation is \[M = \frac{F}{f}\].

Step 2: Substitute Given Values

Substitute the given values for variables M and F into the equation: \[10 = \frac{5}{f}\].

Step 3: Solve for \( f \)

To find \( f \), multiply both sides of the equation by \( f \) to get: \[10f = 5\].

Step 4: Isolate \( f \)

Divide both sides of the equation by 10 to isolate \( f \): \[f = \frac{5}{10}\].

Step 5: Simplify the Fraction

Simplify the fraction: \[f = 0.5\].

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this exercise, we work with an equation that includes fractions. An equation is a statement that two expressions are equal. Understanding how to form and solve equations using algebra is crucial in mathematics. When dealing with equations, we often need to substitute known values for variables and perform operations to isolate the unknown variable.

Variable Isolation

Variable isolation is a key concept in algebra. It involves rearranging an equation to make one variable stand alone on one side of the equation.

Let's break it down:

• First, identify the variable you need to isolate. In our example, it's the variable \( f \).
• Next, use algebraic operations such as addition, subtraction, multiplication, and division to move all other terms to the opposite side of the equation.
• In our case, we multiplied both sides of the equation by \( f \) to remove the fraction: \(10f = 5 \).
• Then, we divided both sides by 10 to isolate \( f \): \( f = \frac{5}{10} \).

By isolating the variable, we find the solution to the equation.

Fractions

Fractions represent a part of a whole and are written in the form of \(\frac{numerator}{denominator}\). In this exercise, fractions were used to express the relationship between the variables.

Here's a simple way to handle fractions in equations:

• To get rid of a fraction, multiply both sides of the equation by the denominator of the fraction.
• In our example, we had \( M = \frac{F}{f} \) which became \(10 = \frac{5}{f} \). By multiplying both sides by the denominator \( f \), we eliminated the fraction.
• We performed a multiplication on both sides: \(10 f = 5 \).
• Once the fraction was cleared, we simplified the equation by isolating \( f \).

Simplifying fractions makes the equation easier to solve. It's essential to practice fraction operations to master solving algebraic equations.