Problem 124 - College Algebra

Question

$$\text { Solve for } y: 3 x+2 y-4=0$$

Step-by-Step Solution

Answer

The solution for 'y' is \(y = 2 - \frac{3x}{2}\).

Step 1: Arrange the Equation

Begin by transferring the term with 'x' and the constant on the other side of the equation: \(3x+2y=4\) . Now, it becomes easier to isolate 'y'.

Step 2: Isolating 'y'

Divide the entire equation by the coefficient of 'y', which is 2. Hence, \(y = \frac{4 - 3x}{2}\).

Step 3: Simplify the Equation

Now, simplify the equation to make 'y' the subject. So, \(y = 2 - \frac{3x}{2}\).

Key Concepts

Isolation of Variables

Solving equations often involves the concept of 'isolation of variables.' The goal is to determine the value of a variable by itself on one side of the equation. Imagine your goal is to pinpoint one friend among a group. Similarly, isolation aims to single out a variable so we can see its direct relationship with other values in the equation.

To isolate a variable, such as 'y' in this context, we take systematic steps to move other elements away from it. These elements can include numbers, constants, or other variables. In the original problem, the term with 'x' and the constant '4' are moved away from 'y' by performing operations that eliminate them from its side of the equation.

Here’s a handy approach to isolating the variable:

• First, decide on the variable you need to isolate, in this case, 'y'.
• Next, perform operations—to both sides—that help shuffle other terms away.
• Keep consistent with operations to maintain equality of the equation.

By leveraging these strategies, you can make 'y' the central focus.

Rearranging Equations

To solve for a specific variable, equations often need to be rearranged. Rearranging simply means changing the form of an equation to facilitate the isolation of variables. It’s similar to cleaning your workspace; sometimes, you need to move things around to really find what you need.

In our example, the goal was to solve for 'y', starting with the equation: \(3x + 2y - 4 = 0\). Recognizing that the left side needs to display 'y' clearly helps us plan our steps. Rearranging involves shifting terms 'around' so each term falls neatly, aiming for clarity and simplicity.

The guideline is to:

• Move terms between sides using addition or subtraction, which keeps the equation balanced
• Tackle any coefficients, by division or multiplication, to properly place the variable as needed

By reorganizing terms, it becomes much easier to perform the final operation of isolation.

Simplification of Expressions

The final key step is the 'simplification of expressions.' Once you've isolated a variable, simplifying means making the equation as neat and understandable as possible. We aim to remove fractions and decimals where feasible and to present the variable explicitly.

Looking at our example, once 'y' was isolated, we achieved: \(y = \frac{4 - 3x}{2}\). However, further simplification is ideal. By dividing out terms, we arrive at:\(y = 2 - \frac{3x}{2}\). Klarity by keeping less clutter.

For simplifying expressions, general tips include:

• Reduce fractions by dividing the numerator and denominator by their greatest common factor (if applicable)
• Combine like terms to minimize complexity

Simplification makes the solution more digestible and practical for anyone working with the equations, staying efficient and clear.