Problem 110 - Intermediate Algebra

Question

Solve \(d_{1} d_{2}=f d_{2}+f d_{1}\) for \(d_{1}\)

Step-by-Step Solution

Answer

\(d_1 = \frac{f d_2}{d_2 - f}\) assuming \(d_2 \neq f\).

Step 1: Understand the Equation

We have the equation \(d_1 d_2 = f d_2 + f d_1\) and need to solve it for \(d_1\). The equation involves two variables, \(d_1\) and \(d_2\), and the constant \(f\).

Step 2: Rearrange the Equation

Firstly, we want to isolate terms involving \(d_1\). To do this, we can write all \(d_1\) terms on one side of the equation:\(d_1 d_2 - f d_1 = f d_2\)

Step 3: Factor Out \(d_1\)

Now, factor out \(d_1\) from the left side of the equation to simplify it:\(d_1 (d_2 - f) = f d_2\).

Step 4: Solve for \(d_1\)

To solve for \(d_1\), divide both sides by \((d_2 - f)\) (assuming \(d_2 eq f\) to avoid division by zero):\(d_1 = \frac{f d_2}{d_2 - f}\).

Key Concepts

Solving Equations

When tackling algebraic equations like \(d_1 d_2 = f d_2 + f d_1\), the goal is often to find the value of one variable in terms of others. Solving equations involves following a systematic approach to simplify and rearrange the terms to isolate the desired variable on one side.

Here are steps to consider when solving an algebraic equation:

• Understand the equation's structure and identify the variables and constants involved.
• Begin by grouping similar terms together to simplify the equation.
• Use operations like addition, subtraction, multiplication, or division to manipulate the equation and isolate the variable of interest.

A key part of solving equations is ensuring that whatever operation is performed on one side of an equation is also performed on the other side. This maintains the balance and equality of the equation. This particular exercise involved recognizing how terms could be rearranged to bring all \(d_1\) terms together, enabling the isolation and solution for \(d_1\).

Factoring

Factoring is an essential skill in algebra that involves breaking down an expression into a product of simpler terms, known as factors. It is a valuable tool when simplifying equations or solving them, especially when terms share common factors.

In this exercise, the equation \(d_1 d_2 - f d_1 = f d_2\) involves factoring out \(d_1\) from the left side. Here’s how it works:

• Notice that both terms \(d_1 d_2\) and \(- f d_1\) include \(d_1\) as a common factor.
• Factor out \(d_1\) from these terms to simplify it into \(d_1 (d_2 - f)\).

Factoring not only simplifies equations but also reveals solutions that may not be immediately apparent. By rewriting terms in a factored form, we can uncover hidden structures within the equation, making the path to solve for the desired variable clear.

Isolating Variables

Isolating the variable is often the end goal when solving an algebraic equation. It means to get the variable by itself on one side of the equation, with everything else on the opposite side. In this exercise, we aimed to solve for \(d_1\).

Here's how you can successfully isolate a variable like \(d_1\):

• Start by simplifying the equation through operations like combining like terms or factoring.
• Once the terms with the variable are on one side, you may need to factor it out, as was done in \(d_1 (d_2 - f) = f d_2\).
• Finally, divide or multiply as necessary to completely isolate the variable. In this case, dividing both sides by \((d_2 - f)\) isolated \(d_1\) on the left, resulting in \(d_1 = \frac{f d_2}{d_2 - f}\).

Paying careful attention to each step and ensuring operations are performed correctly ensures the variable is correctly isolated, allowing for a clear and accurate solution.