3x 5y 15 In Slope Intercept Form

Deconstructing the 3x + 5y = 15 Equation: A Deep Dive into Slope-Intercept Form

The equation 3x + 5y = 15 represents a linear relationship between two variables, x and y. While presented in standard form, its true power and graphical representation become much clearer when transformed into slope-intercept form (y = mx + b). This form reveals the slope (m) and the y-intercept (b) directly, providing crucial insights into the line's characteristics. This article will thoroughly explore the process of converting 3x + 5y = 15 to slope-intercept form, explain the significance of the slope and y-intercept, and delve into various applications and interpretations of this linear equation.

Understanding Standard Form and Slope-Intercept Form

Before diving into the conversion, it's essential to grasp the differences between the two forms:

• Standard Form: Ax + By = C, where A, B, and C are constants, and A is typically non-negative. This form is concise and useful for certain calculations, but it doesn't directly reveal the line's slope or y-intercept.

• Slope-Intercept Form: y = mx + b, where 'm' represents the slope (the steepness of the line) and 'b' represents the y-intercept (the point where the line crosses the y-axis). This form offers immediate visual and analytical insights into the line's behavior.

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

The conversion process involves isolating 'y' on one side of the equation. Let's break it down step-by-step:

1. Subtract 3x from both sides: This moves the 'x' term to the right side of the equation, leaving only the 'y' term on the left. The equation becomes: 5y = -3x + 15

2. Divide both sides by 5: This isolates 'y', giving us the slope-intercept form. The equation becomes: y = (-3/5)x + 3

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

Interpreting the Slope and Y-intercept

With the equation in slope-intercept form, we can readily identify the slope and y-intercept:

• Slope (m) = -3/5: The slope indicates the rate of change of y with respect to x. A negative slope signifies that the line is decreasing (sloping downwards) from left to right. Specifically, for every 5-unit increase in x, y decreases by 3 units.

• Y-intercept (b) = 3: The y-intercept is the point where the line intersects the y-axis. This means when x = 0, y = 3. The coordinates of this point are (0, 3).

Graphical Representation

Plotting the line is straightforward using the slope and y-intercept:

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

2. Use the slope to find another point: The slope is -3/5. From the y-intercept (0, 3), move 5 units to the right (positive x-direction) and 3 units down (negative y-direction). This gives you a second point (5, 0).

3. Draw the line: Draw a straight line passing through these two points (0, 3) and (5, 0). This line represents the graphical representation of the equation 3x + 5y = 15.

Applications and Interpretations

The equation 3x + 5y = 15, and its slope-intercept equivalent, has numerous applications across various fields:

• Economics: This equation could represent a budget constraint, where x and y represent the quantities of two goods, and 15 represents the total budget. The slope (-3/5) indicates the trade-off between the two goods.

• Physics: It could model a relationship between distance (y) and time (x) for an object moving with constant velocity. The slope would represent the velocity, and the y-intercept would be the initial position.

• Engineering: It might describe a linear relationship between two physical quantities in a system. The slope would indicate the proportionality constant between them.

• Data Analysis: If you collect data points that approximately follow a linear trend, you could use linear regression to find the best-fit line, which would be similar to this equation.

Finding the X-intercept

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

0 = (-3/5)x + 3

(3/5)x = 3

x = 5

Therefore, the x-intercept is (5, 0). This confirms our earlier point derived using the slope.

Parallel and Perpendicular Lines

Understanding the slope allows us to determine the equations of lines that are parallel or perpendicular to the given line:

• Parallel Lines: Parallel lines have the same slope. Any line with a slope of -3/5 will be parallel to the line represented by 3x + 5y = 15.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -3/5 is 5/3. Any line with a slope of 5/3 will be perpendicular to the line represented by 3x + 5y = 15.

Solving Systems of Equations

The equation 3x + 5y = 15 can be part of a system of linear equations. Solving such a system involves finding the values of x and y that satisfy both equations simultaneously. Methods like substitution or elimination can be used to find the solution.

Real-World Examples

Let's imagine a few scenarios where this equation might be applicable:

• A fruit stand: Apples (x) cost $3 each, and oranges (y) cost $5 each. You have $15 to spend. The equation 3x + 5y = 15 represents all possible combinations of apples and oranges you can buy with your budget.

• A manufacturing process: The equation could model the relationship between the number of units produced (x) and the total cost (y). The slope reflects the cost per unit, and the y-intercept might represent fixed costs.

• A simple physics problem: An object moves with a constant velocity. The equation could represent the relationship between the distance travelled (y) and the time elapsed (x).

Conclusion

The equation 3x + 5y = 15, when converted to slope-intercept form (y = (-3/5)x + 3), reveals valuable information about the line it represents: its slope (-3/5), its y-intercept (3), and its x-intercept (5). Understanding these elements allows for easy graphical representation, insightful interpretations across various disciplines, and the ability to determine parallel and perpendicular lines. The applications are diverse, ranging from budget constraints in economics to velocity calculations in physics, demonstrating the fundamental importance of linear equations in numerous real-world scenarios. The ability to manipulate and interpret these equations is a crucial skill for anyone working with data or quantitative models.