Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the formula for the specified variable. \(E K+L=D-T K\) for \(K\)

Step-by-Step Solution

Answer

\(K = \frac{D - L}{E + T}\).

Step 1: Move terms involving K to one side

To solve for \(K\), first, we need to bring all the terms containing \(K\) to one side of the equation. Start by adding \(TK\) to both sides to remove it from the right side: \[ EK + TK + L = D \]

Step 2: Isolate the K terms

Next, we need to isolate all terms that contain \(K\). Here, we already have the terms with \(K\) grouped together: \[ (E + T)K + L = D \] Now, subtract \(L\) from both sides to isolate the \(K\) terms: \[ (E + T)K = D - L \]

Step 3: Solve for K

Now that all \(K\) terms are grouped together, divide both sides of the equation by \(E + T\) to solve for \(K\): \[ K = \frac{D - L}{E + T} \]

Key Concepts

Variable Isolation

To find the value of a specific variable, we often aim to isolate it on one side of the equation. This means separating the variable from other terms. In our exercise, we aim to solve for the variable \( K \).

The first step is to gather all terms involving \( K \) on one side. This requires moving the terms using opposite operations. If you encounter subtraction, you add the same value on both sides; if faced with addition, you subtract it instead.

In our example, the term \( TK \) is on the right side of the equation. We need it on the left side, so we add \( TK \) to both sides. This action leaves us with:

• \( EK + TK + L = D \)

Now, \( K \) is only on one side. Remember:

• The key is balancing the equation after each operation.

Algebraic Manipulation

Once you've gathered all the terms involving the variable you're interested in, focus on simplifying the equation as much as possible. This process is called algebraic manipulation, and it's essential for making the equation easier to work with.

Looking at our equation:

• \( (E + T)K + L = D \)

We need to simplify or "manipulate" to help us see \( K \) more clearly. This involves removing any constants or other variables from the same side as \( K \).

For this problem, we subtract \( L \) from both sides to isolate \( K \) terms, transforming the equation into:

• \( (E + T)K = D - L \)

This step highlights all the influences on \( K \) without interference from additional terms. Remember, doing the same operation to both sides keeps the equation balanced.

Equation Rearrangement

Finding the solution often involves rearranging the equation into a form that directly gives us the variable we need. In our problem, the ultimate goal is to solve for \( K \), which involves dividing by the coefficient of \( K \).

In the equation from our previous step:

• \( (E + T)K = D - L \)

The group \( E + T \) is a coefficient with respect to \( K \). We divide both sides by this coefficient to isolate \( K \):

• \( K = \frac{D - L}{E + T} \)

This rearrangement neatly isolates \( K \) on the left. When rearranging equations:

• Check your manipulation steps to ensure each preserves the equation's balance.
• This final step is crucial as it simplifies the answers into their simplest form.