Problem 122 - Precalculus Essentials

Question

Solve for \(y: \quad A x+B y=C y+D\)

Step-by-Step Solution

Answer

So, the solution is \(y = \frac{{D - Ax}}{{B - C}}\)

Step 1: Rearrange the Equation

Begin by grouping all terms with \(y\) together, which involves moving \(Cy\) from the right-hand side of the equation to the left-hand side. This will give: \(Ax + By - Cy = D \)

Step 2: Simplify Equation

Next, simplify the equation by combining like terms (namely, terms that contain \(y\)). This simplifies the equation to: \((B - C)y = D - Ax \)

Step 3: Solve for \(y\)

Finally, to isolate \(y\) as required, divide both sides by \(B - C\). This yields the final answer: \(y = \frac{{D - Ax}}{{B - C}} \)

Key Concepts

Algebraic Manipulation

Algebraic manipulation is like organizing a messy room—you're moving elements around to create order. This involves rearranging terms in an equation to isolate variables, simplify expressions, or make calculations easier. In the example exercise, you started by rearranging the equation to bring all terms containing \(y\) to one side. This is an essential first step. It helps you see your goal more clearly, which is to solve for \(y\).

Imagine you have a set of blocks, each representing part of your equation. Algebraic manipulation lets you move blocks around while ensuring the equation remains balanced. Remember, whatever you do to one side, you must do to the other.

Keep in mind:

• Always perform the same operation on both sides of an equation to keep them equal.
• Move variables on one side to simplify the solving process.
• Pay attention to operation signs when moving terms (e.g., addition changes to subtraction).

Combining Like Terms

Combining like terms makes equations more manageable by turning clusters of similar elements into single entries. It’s like sorting and grouping similar colored marbles. In our equation, you had \(By - Cy\), which needed simplifying. Both terms involved \(y\), making them combinable.

When you combine like terms:

• Sum or subtract coefficients of the same variable.
• This step simplifies expressions and helps clarify what you are solving for.

The transition from \(By - Cy\) to \((B - C)y\) makes the equation much cleaner. It shows a single term with \(y\), simplifying across future steps. This process is crucial for reducing errors and seeing the solution path clearly.

Isolating Variables

Isolating variables is the cornerstone of solving equations. Here, you wanted \(y\) alone on one side of the equation. Isolating the variable involves peeling away layers, like solving a mystery step by step. In this exercise, once like terms were combined, only the step of isolating \(y\) was left.

To isolate \(y\):

• Division or multiplication might be necessary to separate the variable.
• In the given example, you divided both sides by \(B - C\) to solve for \(y\).

This step displays the power of algebraic techniques to unveil what was initially entangled within an equation. Properly isolating a variable ensures clarity and correctness in reaching the solution.