Problem 34 - College Algebra

Question

For the following exercises, solve for the variable. $$ 8(2+4)-15 b=b $$

Step-by-Step Solution

Answer

The solution is \(b = 3\).

Step 1: Expand the Brackets

First, expand the expression inside the brackets in the equation \(8(2+4) - 15b = b\). This means you should calculate \(2+4\) first. The sum is 6, so the equation can be rewritten as \(8 \times 6 - 15b = b\).

Step 2: Simplify Multiplication

Perform the multiplication in the expression \(8 \times 6\). This is equivalent to 48. Substitute 48 back into the equation, resulting in \(48 - 15b = b\).

Step 3: Move Variables to One Side

To solve for \(b\), rearrange the equation so that all terms involving \(b\) are on one side of the equation. Subtract \(b\) from both sides: \(48 - 15b - b = 0\). Simplify the equation to \(48 - 16b = 0\).

Step 4: Solve for the Variable

Isolate \(b\) by adding \(16b\) to both sides: \(48 = 16b\). Now, divide both sides by 16 to solve for \(b\): \(b = \frac{48}{16}\).

Step 5: Simplify the Answer

Simplify \(\frac{48}{16}\) by performing the division. The result is \(b = 3\).

Key Concepts

Equation Simplification

When faced with an equation, the first step is often to simplify it, making it easier to analyze and solve. In this exercise, our equation is given as \(8(2+4) - 15b = b\).

First, recognize that simplification involves resolving any operations in parentheses or brackets. Here, inside the bracket, we have \(2+4\).

By calculating the sum, which is 6, the equation simplifies to \(8 \times 6 - 15b = b\).
Simplification also includes performing basic arithmetic to reduce terms, like calculating \(8 \times 6\), which gives us \(48\).

So, the equation becomes \(48 - 15b = b\).
This practice of breaking down the equation step by step makes further manipulation much simpler. Ultimately, simplification helps in saving time and reducing errors.

Variable Isolation

After simplifying an equation, the next crucial step is isolating the variable, which means getting our variable of interest all by itself on one side of the equation.

In this case, our goal is to solve for \(b\) in the equation \(48 - 15b = b\).

• To do this, we first arrange the terms involving \(b\) on one side. We subtract \(b\) from both sides, resulting in the equation \(48 - 15b - b = 0\).
• This combines to \(48 - 16b = 0\).

At this point, \(b\) is almost isolated, but we still need to get \(b\) completely on its own.

• Add \(16b\) to both sides to move \(-16b\) to the other side, resulting in \(48 = 16b\).
• Finally, divide everything by 16 to isolate \(b\), giving us \(b = \frac{48}{16}\).

Variable isolation is a cornerstone of solving equations, letting us pinpoint exactly what our variable equals.

Basic Arithmetic Operations

The foundation of solving equations is a solid understanding of basic arithmetic operations, which include addition, subtraction, multiplication, and division. These operations are crucial for manipulating equations correctly.

In the exercise at hand, operations are employed to simplify and eventually solve for the variable \(b\). Let's explore:

• First, addition within parentheses \((2+4)\) gives us 6, helping to break down the equation into a more manageable form.
• Next, multiplication \(8 \times 6\) simplifies to 48, effectively removing the parentheses and further clarifying our equation.
• Subtraction is used to combine like terms. Subtract \(b\) from \(-15b\), resulting in \(-16b\).
• Division happens when we need to isolate the variable. Divide by 16 to solve \(b = \frac{48}{16}\), resulting in \(b = 3\).

Mastering these basic operations is essential to solving equations efficiently and accurately.

They ensure that you can combine, reduce, and ultimately isolate variables to find the correct solution.