Problem 32 - Discrete Mathematics with Applications

Question

Find the value of the base \(b\) in each case. $$54 b=64$$

Step-by-Step Solution

Answer

The value of the base \(b\) in the equation \(54 b = 64\) is \(\frac{32}{27}\).

Step 1: Write down the given equation

The given equation is: \(54 b = 64\).

Step 2: Divide both sides of the equation by 54

To solve for $$b$$, we need to get rid of the coefficient $$54$$. To do this, we will divide both sides of the equation by $$54$$. \( \frac{54 b}{54} = \frac{64}{54}\)

Step 3: Simplify both sides of the equation

Once we have divided both sides of the equation, we can simplify it. \(b = \frac{32}{27}\)

Step 4: The value of base $$b$$

Now we have found the value of the base $$b$$. \(b = \frac{32}{27}\) So the value of the base $$b$$ in this equation is \(\frac{32}{27}\).

Key Concepts

Equations

An equation is a mathematical statement that asserts the equality of two expressions. It serves as a bridge that helps us connect and solve different quantitative relationships. In our exercise, the equation is given as \(54b = 64\). This indicates that the product of 54 and an unknown number \(b\) equals 64.

Equations can take many forms, such as linear, quadratic, or polynomial equations. The one we have is a simple linear equation with one variable, which makes it straightforward to solve. Linear equations are the easiest type to handle because they only involve operations of addition, subtraction, multiplication, and division.

Your goal when working with equations is usually to determine the value of the unknown (or variable) that makes the equation true. To do this, you need to perform algebraic manipulations to isolate the variable on one side of the equation. Let's see how this ties into solving equations.

Solving equations

Solving an equation involves finding the value of the variable that makes the equation true. In our original exercise, we aim to find the value of \(b\) in the equation \(54b = 64\). The process of solving equations typically involves the following steps:

1. **Identify the equation:** Start by recognizing and writing down the equation. Here, it's \(54b = 64\).
2. **Isolate the variable:** Next, modify the equation to get the variable on its own. In this example, we do this by dividing both sides by 54, leading to \(b = \frac{64}{54}\).
3. **Simplify:** Once the variable is isolated, simplify the equation if possible. From \(b = \frac{64}{54}\), we further simplify to \(b = \frac{32}{27}\).

This process can vary slightly depending on the type of equation. However, isolating the variable remains a consistent principle in solving most equations. Practice makes perfect, so try solving more equations to get comfortable with these steps.

Rational numbers

A rational number is any number that can be expressed as the quotient or fraction \(\frac{a}{b}\) of two integers, with the denominator \(b\) not equal to zero. Rational numbers include integers, fractions, and finite decimals.

In our exercise, the solution for \(b\) is a rational number: \(\frac{32}{27}\). Here's how you know it's a rational number:

• It is written as a fraction.
• Both 32 and 27 are integers.
• The denominator 27 is not zero.

Rational numbers are significant in algebra because they allow us to express exact values that are not integers. Recognizing and understanding how to work with these numbers is essential, especially when simplifying equations or performing arithmetic operations.

Remember, in algebra, solutions are often best presented as rational numbers, especially when dealing with fractions and decimals. This helps in maintaining precision and clarity, as seen in the solution to our exercise.