Solving for y in 3x 2y 8: A Comprehensive Guide

When faced with the algebraic equation 3x 2y 8, finding the value of y in terms of x can provide valuable insight into the relationship between these two variables. Let's explore the step-by-step process to solve for y and understand the key algebraic concepts involved.

Step-by-Step Guide to Solving for y

To solve for y in the equation 3x 2y 8, follow these detailed steps:

Isolate the term with y by subtracting 3x from both sides of the equation:

3x 2y 8

2y 8 - 3x

Divide both sides by 2 to solve for y:

y left( frac{8 - 3x}{2} right)

Simplify the expression:

y 4 - frac{3}{2}x

Visualizing the Solution

The solution shows that the relationship between x and y is linear. This can be better understood by plotting the points on a graph. Let's take a look at some specific examples to see how these points form a straight line.

Example Solutions for (x, y)

Here are several examples of specific (x, y) values satisfying the equation:

For x 2, y 1 For x 4, y -2 For x 6, y -5 For x 8, y -8

Note: Each pair (x, y) satisfies the equation 3x 2y 8 and can be plotted on a graph to visualize the straight line that represents this linear relationship.

Alternative Methods to Solve for y

Let's explore alternative methods to solve for y in 3x 2y 8:

Add -3x on both sides:

3x 2y - 3x 8 - 3x

2y 8 - 3x

left( frac{2y}{2} right) left( frac{8 - 3x}{2} right)

y 4 - left( frac{3}{2}x right)

Subtract 3x from both sides:

2y 8 - 3x

left( frac{2y}{2} right) left( frac{8 - 3x}{2} right)

y 4 - left( frac{3}{2}x right)

Subtract 3x from both sides:

2y -3x 8

y -left( frac{3}{2}x right) 4

Subtract 3x from both sides:

2y 8 - 3x

left( frac{2y}{2} right) left( frac{8 - 3x}{2} right)

y 4 - left( frac{3}{2}x right)

Note: These methods all lead to the same simplified equation, demonstrating the consistency and reliability of the algebraic approach.

Visual Representation of the Solution

The equation 3x 2y 8 can be graphed as a straight line, reflecting the linear relationship between x and y. To visualize this, plot the equation using a graphing tool such as Wolfram Alpha. This graph will confirm the linear nature of the relationship and provide a visual representation of the solutions for (x, y).

In conclusion, solving for y in the equation 3x 2y 8 through algebraic manipulation confirms a linear relationship. Understanding and applying these steps to solve for y demonstrates the fundamental skills in algebra and graphing that are essential in various mathematical and scientific contexts.