5x 4y 8 In Slope Intercept Form

Decoding the Equation: 5x + 4y = 8 in Slope-Intercept Form

The equation 5x + 4y = 8 represents a linear relationship between two variables, x and y. While presented in standard form, understanding its slope-intercept form (y = mx + b, where 'm' is the slope and 'b' is the y-intercept) provides crucial insights into the line's characteristics and behavior. This comprehensive guide will explore the transformation process, interpret the results, and delve into related concepts, equipping you with a strong grasp of linear equations.

Understanding the Standard Form and Slope-Intercept Form

Before we embark on the transformation, let's clarify the significance of both forms:

Standard Form: Ax + By = C

The standard form, Ax + By = C, where A, B, and C are constants, provides a concise representation of a linear equation. However, it doesn't directly reveal the slope and y-intercept, crucial elements for visualizing and understanding the line's properties.

Slope-Intercept Form: y = mx + b

The slope-intercept form, y = mx + b, explicitly presents the line's slope (m) and y-intercept (b). The slope indicates the steepness and direction of the line (positive slope implies an upward trend, negative slope a downward trend), while the y-intercept represents the point where the line crosses the y-axis (the point where x = 0). This form is particularly useful for graphing the line quickly and intuitively.

Transforming 5x + 4y = 8 into Slope-Intercept Form

The goal is to isolate 'y' on one side of the equation, thereby revealing the slope and y-intercept. Here's a step-by-step approach:

1. Subtract 5x from both sides: This step moves the 'x' term to the right-hand side, leaving only the 'y' term on the left. 4y = -5x + 8

2. Divide both sides by 4: This isolates 'y', giving us the slope-intercept form. y = (-5/4)x + 2

Interpreting the Results: Slope and Y-Intercept

Now that we have the equation in slope-intercept form, y = (-5/4)x + 2, we can readily identify the slope and y-intercept:

Slope (m) = -5/4

The slope of -5/4 indicates that for every 4 units increase in the x-value, the y-value decreases by 5 units. This signifies a negative slope, meaning the line descends from left to right.

Y-Intercept (b) = 2

The y-intercept of 2 indicates that the line intersects the y-axis at the point (0, 2). This is the point where x = 0.

Graphing the Line

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

1. Plot the y-intercept: Mark the point (0, 2) on the y-axis.

2. Use the slope to find another point: Since the slope is -5/4, move 4 units to the right and 5 units down from the y-intercept. This gives you the point (4, -3).

3. Draw the line: Connect the two points (0, 2) and (4, -3) with a straight line. This line represents the equation 5x + 4y = 8.

Finding the X-Intercept

While the y-intercept is readily obtained from the slope-intercept form, the x-intercept (the point where the line crosses the x-axis, where y = 0) can be found by setting y = 0 in the original equation or the slope-intercept form:

Using the original equation: 5x + 4(0) = 8 => 5x = 8 => x = 8/5 = 1.6

Using the slope-intercept form: 0 = (-5/4)x + 2 => (5/4)x = 2 => x = (4/5) * 2 = 8/5 = 1.6

Therefore, the x-intercept is (1.6, 0).

Applications and Real-World Examples

Linear equations, particularly in slope-intercept form, find wide applications in various fields:

• Economics: Modeling supply and demand, predicting economic growth.
• Physics: Representing velocity and acceleration, analyzing projectile motion.
• Engineering: Designing structures, calculating fluid flow.
• Computer Science: Developing algorithms, creating graphical representations.

Example: Imagine a phone plan with a fixed monthly fee and a per-minute charge. The total cost (y) can be modeled as a linear equation where the slope represents the per-minute charge and the y-intercept represents the fixed monthly fee.

Advanced Concepts and Extensions

This foundational understanding of linear equations can be extended to more complex scenarios:

• Systems of Linear Equations: Solving multiple linear equations simultaneously to find intersection points.
• Linear Inequalities: Graphing regions representing inequalities rather than just lines.
• Linear Programming: Optimizing linear objective functions subject to linear constraints.
• Multivariate Linear Regression: Extending the concepts to multiple variables for predictive modeling.

Conclusion

Transforming the equation 5x + 4y = 8 into slope-intercept form (y = (-5/4)x + 2) reveals essential information about the line it represents: its slope (-5/4) indicating its downward trend, and its y-intercept (2) showing its crossing point on the y-axis. This form simplifies graphing and provides valuable insights for various applications. Understanding this process lays a strong foundation for more advanced concepts in linear algebra and its applications in diverse fields. Mastering this fundamental transformation empowers you to analyze and interpret linear relationships effectively. By understanding the slope and y-intercept, you unlock a deeper understanding of the line's behavior and its implications within broader mathematical and real-world contexts. The ability to move fluidly between standard form and slope-intercept form is a key skill in algebra, showcasing your analytical prowess and problem-solving capabilities.