5x 3y 15 In Slope Intercept Form

Understanding and Applying the Slope-Intercept Form: Solving 5x - 3y = 15

The equation 5x - 3y = 15 represents a linear relationship between two variables, x and y. Understanding this relationship and expressing it in the slope-intercept form, y = mx + b, is crucial in algebra and numerous applications. This comprehensive guide will not only show you how to convert the given equation but also delve into the meaning of slope and y-intercept, explore various applications, and answer frequently asked questions.

Introduction: What is Slope-Intercept Form?

The slope-intercept form of a linear equation, y = mx + b, is a powerful tool for visualizing and understanding straight lines. In this equation:

• m represents the slope of the line. 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 indicates a line that falls from left to right. 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). It's the y-coordinate of the point (0, b).

Converting equations into slope-intercept form allows us to easily identify the slope and y-intercept, which provide valuable insights into the line's characteristics and behavior.

Steps to Convert 5x - 3y = 15 to Slope-Intercept Form

Our goal is to manipulate the equation 5x - 3y = 15 to isolate y and express it in the form y = mx + b. Here's a step-by-step guide:

1. Subtract 5x from both sides: This step aims to move the term containing x to the right-hand side of the equation. The result is:

   -3y = -5x + 15

2. Divide both sides by -3: This isolates y, giving us the slope-intercept form. Remember to divide every term on both sides by -3:

   y = (5/3)x - 5

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

Interpreting the Slope and Y-Intercept

From the equation y = (5/3)x - 5, we can easily identify:

• Slope (m) = 5/3: This positive slope indicates that the line rises from left to right. For every 3 units increase in x, y increases by 5 units.

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

Graphical Representation

Plotting the line represented by y = (5/3)x - 5 is straightforward. We already know one point: the y-intercept (0, -5). To find another point, we can choose any value for x and calculate the corresponding y value. For example, if we let x = 3:

y = (5/3)(3) - 5 = 5 - 5 = 0

This gives us the point (3, 0). Plotting these two points (0, -5) and (3, 0) and drawing a straight line through them will represent the equation 5x - 3y = 15 graphically.

Further Applications and Extensions

The slope-intercept form has numerous applications beyond basic algebra. Here are some examples:

• Modeling real-world phenomena: Linear equations can model various real-world situations, such as the relationship between distance and time, cost and quantity, or temperature and altitude. The slope and y-intercept then have meaningful interpretations within the context of the problem. For example, a linear equation could model the cost of a taxi ride, where the slope represents the cost per mile and the y-intercept represents the initial fare.

• Predicting values: Once you have a linear equation in slope-intercept form, you can easily predict the value of y for any given value of x, or vice-versa. This is valuable for forecasting and making estimations.

• Finding parallel and perpendicular lines: Parallel lines have the same slope but different y-intercepts. Perpendicular lines have slopes that are negative reciprocals of each other (e.g., if one line has a slope of 2, a perpendicular line would have a slope of -1/2). Understanding slope allows us to easily determine the relationships between lines.

• Linear Programming: In operations research, linear programming utilizes linear equations to optimize resource allocation and decision-making processes. The slope-intercept form is frequently employed in graphical methods for solving linear programming problems.

Explanation of the Mathematical Concepts Involved

The conversion process from the standard form (Ax + By = C) to the slope-intercept form (y = mx + b) involves fundamental algebraic manipulations. Understanding these manipulations is critical for grasping the underlying principles.

• Solving for a variable: The core of the conversion is isolating the variable 'y' on one side of the equation. This is achieved through a sequence of operations, including addition, subtraction, multiplication, and division, ensuring that whatever operation is performed on one side of the equation is also done on the other to maintain equality.

• The concept of equivalence: Each step in the conversion process maintains the equivalence of the equation. The equation remains unchanged in terms of its representation of the line, only its form is altered for easier interpretation and application.

• The role of the slope and y-intercept: The slope and y-intercept are parameters that completely define a unique straight line in a two-dimensional coordinate system. Their values are extracted directly from the slope-intercept form, making it a convenient representation for understanding and manipulating linear relationships.

Frequently Asked Questions (FAQ)

• Q: What if the equation is not in the standard form Ax + By = C?

  A: If the equation is in a different form, you'll need to first manipulate it algebraically into the standard form before converting to the slope-intercept form.

• Q: Can a vertical line be expressed in slope-intercept form?

  A: No. A vertical line has an undefined slope (because the change in x is zero), making it impossible to express in the y = mx + b form. Vertical lines are typically represented by the equation x = c, where c is a constant.

• Q: Can a horizontal line be expressed in slope-intercept form?

  A: Yes. A horizontal line has a slope of 0, and its equation can be written as y = b, where b is the y-intercept.

• Q: What are some common mistakes to avoid when converting to slope-intercept form?

  A: Common mistakes include: forgetting to divide all terms by the coefficient of y, making errors in arithmetic (especially with signs), and misunderstanding the concept of slope and y-intercept. Careful attention to detail and a methodical approach are essential to avoid these errors.

Conclusion: Mastering the Slope-Intercept Form

The slope-intercept form, y = mx + b, is an indispensable tool in algebra and various applications. Converting equations into this form provides valuable insights into the characteristics of a linear relationship, its graphical representation, and its use in problem-solving. By understanding the concepts of slope and y-intercept, and by mastering the algebraic manipulations involved in the conversion process, you'll build a solid foundation for further mathematical explorations. Remember to practice regularly to solidify your understanding and build confidence in applying this crucial concept. The conversion of 5x - 3y = 15 to y = (5/3)x - 5 is a straightforward example that illustrates the power and simplicity of the slope-intercept form.