Problem 14 - College Algebra

Question

For each of the following exercises, solve the equation for \(y\) in terms of \(x\) . $$x-2 y=7$$

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = \frac{x - 7}{2}\).

Step 1: Understand the Equation

The given equation is linear and written in the form \(x - 2y = 7\). The goal is to solve for \(y\) in terms of \(x\). This means we'll rearrange the equation to express \(y\) on one side and \(x\) on the other side.

Step 2: Isolate the y-term

To isolate the \(y\)-term, subtract \(x\) from both sides of the equation:\[ -2y = 7 - x \] This operation aims to have all \(y\)-related terms on one side.

Step 3: Solve for y

Now divide every term by \(-2\) to solve for \(y\):\[ y = \frac{7 - x}{-2} \]Rearrange the equation to express \(y\) in a more conventional format as:\[ y = \frac{-7 + x}{2} \] or \[ y = \frac{x - 7}{2} \]

Key Concepts

Solving for y

To solve the equation for \( y \) in terms of \( x \), we're focusing on getting \( y \) by itself on one side of the equation. This type of task is common in algebra when we need to express one variable in terms of another.
It's all about breaking down the operations step-by-step. We carefully apply inverse operations like addition, subtraction, multiplication, or division to isolate the variable \( y \).
Let's consider our example equation:

• Start with the given equation: \( x - 2y = 7 \).
• Our target is to manipulate this equation to finally express \( y \) alone on one side.

Solving for \( y \) is akin to peeling an onion layer by layer. At each step, we apply an algebraic operation that helps strip away the other variables or numbers. The goal is to simplify the equation until \( y \) stands alone.

Equation Manipulation

When handling equations, manipulation refers to the process of performing various operations to alter the equation's structure without changing its equality. Especially with linear equations, we manipulate to rearrange terms.
In our exercise, we have:

• Start with: \( x - 2y = 7 \)
• Perform the same operation to both sides, such as subtracting \( x \), which gives us \(-2y = 7 - x\).

This step is about maintaining balance and equality. Each modification or operation on one side must be mirrored on the other, akin to keeping a seesaw balanced.
Effective equation manipulation involves understanding how each operation affects the equation as a whole. As we see in our example, this strategic manipulation helps to bring the equation closer to our desired outcome, where \( y \) is isolated.

Rearranging terms

Rearranging terms in an equation is like tidying up a room. We strategically move pieces around to achieve a clutter-free presentation. The aim is to enhance clarity and purposefulness.
When solving equations for a specific variable, one of the tasks is to ensure the required variable is isolated. This involves:

• Grouping all terms involving the variable \( y \) on one side.
• Shifting constants and terms involving other variables to the opposite side.

In our example, after the equation manipulation, we had \(-2y = 7 - x\).
Rearranging involves simplifying it further by dividing everything by \(-2\), resulting in \( y = \frac{7 - x}{-2} \). Then, reordering the terms gives a more familiar representation: \( y = \frac{x - 7}{2} \).
The ultimate aim of rearranging is to simplify and solve the equation by making the variable of interest, \( y \), the focus. It's about making the output as simple and straight-forward as possible for better interpretation.