5x 3y 15 In Slope Intercept Form

5x + 3y = 15 in Slope-Intercept Form: A Comprehensive Guide

The equation 5x + 3y = 15 represents a linear relationship between two variables, x and y. While useful in its current form, converting it to slope-intercept form (y = mx + b) provides valuable insights into the line's characteristics – specifically its slope (m) and y-intercept (b). This guide will delve deep into the process of this conversion, explore the significance of the slope and y-intercept, and demonstrate how to graph the line. We'll also examine related concepts and provide practical applications to solidify your understanding.

Understanding Slope-Intercept Form (y = mx + b)

Before diving into the conversion, let's solidify our understanding of the slope-intercept form, y = mx + b. This form is incredibly useful because:

• m represents the slope: The slope indicates the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope means it falls. The magnitude of the slope indicates the steepness; a larger absolute value signifies a steeper line.

• b represents the y-intercept: The y-intercept is the point where the line crosses the y-axis. It's the value of y when x is 0.

Converting 5x + 3y = 15 to Slope-Intercept Form

The key to converting 5x + 3y = 15 to slope-intercept form is to isolate 'y' on one side of the equation. We'll achieve this through a series of algebraic manipulations:

1. Subtract 5x from both sides: This moves the x term to the right side of the equation, resulting in:

   3y = -5x + 15

2. Divide both sides by 3: This isolates 'y', giving us the slope-intercept form:

   y = (-5/3)x + 5

Now we have our equation in slope-intercept form: y = (-5/3)x + 5. This tells us immediately:

• Slope (m) = -5/3: The line has a negative slope, meaning it slopes downwards from left to right. The slope of -5/3 indicates that for every 3 units moved to the right along the x-axis, the line falls 5 units along the y-axis.

• Y-intercept (b) = 5: The line intersects the y-axis at the point (0, 5).

Graphing the Line

With the slope and y-intercept in hand, graphing the line y = (-5/3)x + 5 is straightforward:

1. Plot the y-intercept: Begin by plotting the point (0, 5) on the y-axis.

2. Use the slope to find another point: The slope is -5/3. This can be interpreted as a rise of -5 and a run of 3. Starting from the y-intercept (0,5), move 3 units to the right (run) and 5 units down (rise) to locate another point on the line. This point would be (3, 0). You could also move 3 units to the left and 5 units up to find another point (-3,10).

3. Draw the line: Draw a straight line passing through the two points you've plotted. This line represents the equation 5x + 3y = 15.

Further Exploration: Finding the x-intercept

While the y-intercept is readily available from the slope-intercept form, the x-intercept (where the line crosses the x-axis) can be found by setting y = 0 and solving for x:

0 = (-5/3)x + 5

(5/3)x = 5

x = 3

Therefore, the x-intercept is (3, 0). This confirms the point we found using the slope.

Practical Applications

Understanding linear equations and their slope-intercept form has wide-ranging applications across various fields:

• Economics: Supply and demand curves are often represented by linear equations. The slope represents the responsiveness of quantity demanded or supplied to changes in price.

• Physics: Many physical phenomena can be modeled using linear equations. For example, the relationship between distance, speed, and time.

• Engineering: Linear equations are crucial in structural analysis, electrical circuits, and many other engineering disciplines.

• Data Analysis: Linear regression, a fundamental statistical technique, relies on fitting a linear equation to a set of data points. The slope and intercept provide insights into the relationship between the variables.

• Computer Graphics: Lines are fundamental elements in computer graphics, and understanding their equations is essential for creating and manipulating images.

Variations and Extensions

Let's consider variations on the original equation and explore how they impact the slope and intercept:

• 5x + 3y = 30: This equation represents a parallel line to y = (-5/3)x + 5, but with a y-intercept of 10. The slope remains the same (-5/3). Parallel lines have the same slope.

• -5x - 3y = 15: This equation represents a line with the same slope as the original, but with a different y-intercept. Simplifying to slope-intercept form will yield y = (-5/3)x -5.

• 10x + 6y = 30: This equation, when simplified, will result in the same slope-intercept form as the original equation, y = (-5/3)x + 5. This demonstrates that equivalent equations can be represented in different forms.

Solving Problems involving Parallel and Perpendicular Lines

Understanding the slope is vital when dealing with parallel and perpendicular lines:

• Parallel Lines: Parallel lines have the same slope but different y-intercepts. If you are given a line and asked to find the equation of a parallel line passing through a specific point, you know the slope will be the same. You can then use the point-slope form of a line (y - y1 = m(x - x1)) to find the equation.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. If the slope of one line is 'm', the slope of a perpendicular line will be '-1/m'. Again, the point-slope form is useful for determining the equation of a perpendicular line passing through a specific point.

Conclusion

Converting the equation 5x + 3y = 15 to slope-intercept form (y = (-5/3)x + 5) reveals valuable information about the line's characteristics: its slope (-5/3) and its y-intercept (5). Understanding these concepts, along with the ability to graph the line and solve related problems, is fundamental to a solid grasp of linear algebra and its various applications. This comprehensive guide provides a solid foundation for further exploration of linear equations and their multifaceted uses in different fields. Remember to practice regularly to solidify your understanding and build confidence in tackling more complex linear equations.