3x Y 2 In Slope Intercept Form

Deconstructing the Equation 3x + 2y = 6: A Deep Dive into Slope-Intercept Form

The equation 3x + 2y = 6 represents a linear relationship between two variables, x and y. While presented in standard form, its true power and ease of interpretation become apparent when transformed into slope-intercept form: y = mx + b. This form reveals the line's slope (m) and y-intercept (b) instantly, providing valuable insights into the line's characteristics and behavior. This article will dissect the transformation process, explore the meaning of slope and y-intercept, and illustrate how this understanding unlocks a deeper comprehension of linear equations.

From Standard Form to Slope-Intercept Form: The Transformation

The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. Our equation, 3x + 2y = 6, fits this mold perfectly. To convert it to slope-intercept form (y = mx + b), we need to isolate 'y' on one side of the equation. Let's walk through the steps:

1. Subtract 3x from both sides: This isolates the term containing 'y'. The equation becomes: 2y = -3x + 6

2. Divide both sides by 2: This isolates 'y' completely. The equation now reads: y = (-3/2)x + 3

Now, we have our equation in slope-intercept form: y = (-3/2)x + 3.

Understanding the Slope (m) and the Y-Intercept (b)

The transformed equation reveals two crucial pieces of information:

• Slope (m) = -3/2: The slope represents the steepness and direction of the line. It's the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A negative slope, like in this case, indicates a line that slopes downwards from left to right. The numerical value, 3/2, means that for every 2 units moved horizontally to the right, the line moves 3 units downwards.

• Y-intercept (b) = 3: The y-intercept is the point where the line intersects the y-axis (where x = 0). In our equation, the y-intercept is 3, meaning the line passes through the point (0, 3).

Visualizing the Line: Graphing the Equation

Now that we understand the slope and y-intercept, we can easily graph the line represented by the equation y = (-3/2)x + 3.

1. Plot the y-intercept: Start by plotting the point (0, 3) on the coordinate plane.

2. Use the slope to find another point: The slope is -3/2. From the y-intercept (0, 3), move 2 units to the right (horizontal run) and 3 units down (vertical rise). This brings us to the point (2, 0).

3. Draw the line: Draw a straight line passing through the two points (0, 3) and (2, 0). This line represents the equation 3x + 2y = 6.

Applications and Extensions: Real-World Significance

Understanding linear equations in slope-intercept form has far-reaching applications in various fields:

• Physics: Describing the motion of objects with constant velocity. The slope represents the velocity, and the y-intercept represents the initial position.

• Economics: Modeling supply and demand curves. The slope represents the change in quantity demanded or supplied in response to a change in price.

• Engineering: Representing relationships between physical quantities, like voltage and current in a circuit.

• Computer Science: Representing linear transformations in computer graphics and image processing.

Solving Problems Using the Slope-Intercept Form

The slope-intercept form makes solving various problems related to the line incredibly straightforward. Let's explore some examples:

1. Finding the y-coordinate given an x-coordinate:

Suppose we want to find the y-coordinate when x = 4. We simply substitute x = 4 into the equation:

y = (-3/2)(4) + 3 = -6 + 3 = -3

Therefore, the point (4, -3) lies on the line.

2. Finding the x-coordinate given a y-coordinate:

Let's find the x-coordinate when y = 0. We substitute y = 0 into the equation and solve for x:

0 = (-3/2)x + 3

(3/2)x = 3

x = 2

This confirms that the x-intercept is indeed 2, corresponding to the point (2,0).

3. Determining Parallel and Perpendicular Lines:

• Parallel Lines: Parallel lines have the same slope. Any line with a slope of -3/2 will be parallel to our line.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -3/2 is 2/3. Any line with a slope of 2/3 will be perpendicular to our line.

Advanced Concepts and Extensions

While the basic concepts of slope and y-intercept are fundamental, understanding them unlocks the ability to delve into more complex topics:

• Systems of Linear Equations: Solving systems of linear equations often involves graphing the lines represented by the equations and finding their point of intersection. The slope-intercept form makes visualizing and solving these systems much easier.

• Linear Inequalities: Extending the concept to linear inequalities involves shading regions on the coordinate plane that satisfy the inequality. The slope-intercept form provides a clear visual representation for these inequalities.

• Linear Regression: In statistics, linear regression aims to find the line of best fit for a set of data points. Understanding slope and intercept is crucial for interpreting the results of a linear regression analysis.

Conclusion: Mastering the Slope-Intercept Form

The transformation of the equation 3x + 2y = 6 into slope-intercept form, y = (-3/2)x + 3, unveils a wealth of information about the line it represents. The slope (-3/2) and y-intercept (3) provide a clear picture of the line's steepness, direction, and intersection with the y-axis. This understanding extends beyond simple graphing, enabling the solving of various problems and paving the way for deeper exploration into more advanced mathematical concepts. Mastering the slope-intercept form is not just about manipulating equations; it's about gaining a profound understanding of linear relationships and their diverse applications in the real world. By understanding this fundamental concept, you equip yourself with a powerful tool for solving problems across numerous fields.