3x Y 6 In Slope Intercept Form

Understanding and Converting 3x + y = 6 to Slope-Intercept Form

The equation 3x + y = 6 represents a linear relationship between two variables, x and y. While this equation is in standard form (Ax + By = C), it's often more useful to express it in slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. This form allows for easy visualization of the line's characteristics and simplifies various calculations. This article will comprehensively guide you through the process of converting 3x + y = 6 into slope-intercept form and explore the implications of this transformation.

What is Slope-Intercept Form?

Before delving into the conversion, let's solidify our understanding of the slope-intercept form: y = mx + b.

• y: Represents the dependent variable, typically plotted on the vertical axis of a graph.
• x: Represents the independent variable, typically plotted on the horizontal axis.
• m: Represents the slope of the line. The slope indicates the steepness and direction of the line. A positive slope means the line ascends from left to right, while a negative slope means it descends. The slope is calculated as the change in y divided by the change in x (rise over run).
• b: Represents the y-intercept. This is the point where the line intersects the y-axis (where x = 0).

Converting 3x + y = 6 to Slope-Intercept Form (y = mx + b)

The conversion process involves isolating 'y' on one side of the equation. Let's break down the steps:

1. Start with the original equation: 3x + y = 6

2. Subtract 3x from both sides: This step aims to move the 'x' term to the right side of the equation. Subtracting 3x from both sides maintains the equation's balance. The result is:

   y = -3x + 6

Now, we have successfully converted the equation into slope-intercept form (y = mx + b).

Analyzing the Slope and Y-Intercept

Having successfully transformed the equation, we can now extract valuable information about the line it represents:

• Slope (m) = -3: This indicates a negative slope. The line will descend from left to right. For every one-unit increase in x, y will decrease by three units.

• Y-intercept (b) = 6: This means the line intersects the y-axis at the point (0, 6).

Graphing the Line

With the slope and y-intercept readily available, graphing the line becomes straightforward.

1. Plot the y-intercept: Begin by plotting the point (0, 6) on the y-axis.

2. Use the slope to find another point: The slope is -3, which can be expressed as -3/1. This means a rise of -3 and a run of 1. Starting from the y-intercept (0, 6), move down 3 units (because of the -3 rise) and then 1 unit to the right (because of the 1 run). This will bring you to the point (1, 3).

3. Draw the line: Draw a straight line that passes through both points (0, 6) and (1, 3). This line represents the equation 3x + y = 6.

Practical Applications and Further Exploration

Understanding the slope-intercept form has far-reaching applications in various fields, including:

• Economics: Analyzing linear relationships between variables like price and demand.
• Physics: Modeling the motion of objects with constant acceleration.
• Engineering: Designing structures and systems.
• Computer Science: Creating algorithms and simulations involving linear relationships.

Furthermore, exploring variations of the equation 3x + y = 6 provides additional learning opportunities:

Finding the x-intercept

The x-intercept is the point where the line crosses the x-axis (where y = 0). To find it, substitute y = 0 into the original equation (or the slope-intercept form):

3x + 0 = 6 3x = 6 x = 2

Therefore, the x-intercept is (2, 0).

Parallel and Perpendicular Lines

The slope plays a crucial role in determining the relationship between lines:

• Parallel Lines: Parallel lines have the same slope. Any line parallel to 3x + y = 6 will also have a slope of -3.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -3 is 1/3. Any line perpendicular to 3x + y = 6 will have a slope of 1/3.

Real-World Examples

Let's consider a few scenarios where understanding the slope-intercept form is beneficial:

• Calculating the cost of a phone plan: A phone plan might cost $10 per month plus $0.10 per minute. This can be represented as y = 0.10x + 10, where y is the total cost, x is the number of minutes, the slope (0.10) represents the cost per minute, and the y-intercept (10) represents the fixed monthly cost.

• Tracking the growth of a plant: If a plant grows at a consistent rate of 2 cm per week and starts at a height of 5 cm, this can be expressed as y = 2x + 5, where y is the height in centimeters and x is the number of weeks.

Conclusion

Converting the equation 3x + y = 6 to slope-intercept form (y = -3x + 6) reveals critical information about the linear relationship it represents: a slope of -3 and a y-intercept of 6. This form simplifies graphing, analyzing, and applying this equation in various contexts. Understanding the concept of slope and y-intercept empowers you to interpret linear relationships effectively and apply them to practical problems across diverse fields. The ability to manipulate and interpret linear equations is a fundamental skill in mathematics and has far-reaching implications in numerous fields of study and professional practices. Remember to practice regularly to solidify your understanding and build confidence in your ability to handle similar conversions and applications.