5x 3y 9 In Slope Intercept Form

Converting 5x + 3y = 9 to Slope-Intercept Form: A Comprehensive Guide

The equation 5x + 3y = 9 represents a linear relationship between two variables, x and y. However, it's not in a format that readily reveals key characteristics like the slope and y-intercept. To gain a clearer understanding of this line, we need to convert it into slope-intercept form, which is typically written as y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. This guide will walk you through the process step-by-step, explaining the underlying concepts and providing practical examples.

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

Before diving into the conversion, let's solidify our understanding of the slope-intercept form:

y: Represents the dependent variable. Its value depends on the value of x.
x: Represents the independent variable. Its value is freely chosen.
m: Represents the slope of the line. The slope describes the steepness and direction of the line. A positive slope indicates an upward trend (from left to right), while a negative slope indicates a downward trend. 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 crosses the y-axis (where x = 0).

Converting 5x + 3y = 9 to Slope-Intercept Form

The goal is to isolate 'y' on one side of the equation, leaving the equation in the form y = mx + b. Let's break down the conversion process:

Subtract 5x from both sides:

Our starting equation is: 5x + 3y = 9

Subtracting 5x from both sides gives us: 3y = -5x + 9
Divide both sides by 3:

To isolate 'y', we divide both sides of the equation by 3:

(3y)/3 = (-5x + 9)/3

This simplifies to: y = (-5/3)x + 3
Identify the slope and y-intercept:

Now that the equation is in slope-intercept form (y = mx + b), we can easily identify:
- Slope (m) = -5/3: This indicates a negative slope, meaning the line slopes downwards from left to right. The slope of -5/3 means that for every 3 units of increase in x, y decreases by 5 units.
- y-intercept (b) = 3: This means the line crosses the y-axis at the point (0, 3).

Graphical Representation

Plotting the line on a graph provides a visual representation of the equation. Using the y-intercept (0, 3) as a starting point, and the slope (-5/3), we can easily plot additional points:

Start at (0, 3): This is our y-intercept.
Use the slope to find another point: The slope is -5/3. This means a movement of 3 units to the right and 5 units down will give another point on the line. Starting at (0, 3), move 3 units to the right (+3 to the x-coordinate) and 5 units down (-5 to the y-coordinate), giving us the point (3, -2).
Plot the points and draw the line: Plot the points (0, 3) and (3, -2) on a graph. Draw a straight line through these points, extending it in both directions. This line represents the equation y = (-5/3)x + 3.

Applications and Further Exploration

Understanding the slope-intercept form of a linear equation has numerous applications in various fields:

Economics: Analyzing relationships between supply and demand, cost and revenue.
Physics: Studying motion, velocity, and acceleration.
Engineering: Modeling linear relationships in various systems.
Statistics: Analyzing linear correlations between variables.
Computer Science: In algorithms and data structures where linear relationships are important.

Deeper Dive: Different Forms of Linear Equations

While the slope-intercept form is very useful, it's not the only way to represent a linear equation. Other common forms include:

Standard Form (Ax + By = C): This form emphasizes the coefficients of x and y and the constant term. Our original equation, 5x + 3y = 9, is in standard form.
Point-Slope Form (y - y₁ = m(x - x₁)): This form is useful when you know the slope and a point on the line.

Converting between these forms is a valuable skill for solving various mathematical problems. For example, you can easily convert the slope-intercept form back to the standard form by multiplying both sides by 3 and rearranging the terms.

Practical Examples and Problem Solving

Let's explore some practical examples to solidify our understanding:

Example 1:

Find the equation of a line with a slope of 2 and a y-intercept of -1.

Solution: Using the slope-intercept form, y = mx + b, we substitute m = 2 and b = -1. The equation is y = 2x - 1.

Example 2:

Find the slope and y-intercept of the line represented by the equation 2x - 4y = 8.

Solution: Convert the equation to slope-intercept form:
1. Subtract 2x from both sides: -4y = -2x + 8
2. Divide both sides by -4: y = (1/2)x - 2
Therefore, the slope (m) is 1/2, and the y-intercept (b) is -2.

Example 3:

Determine if the points (1, 2) and (4, 5) lie on the line y = x + 1.

Solution: Substitute the x and y values of each point into the equation:
- Point (1, 2): 2 = 1 + 1 (True)
- Point (4, 5): 5 = 4 + 1 (True)
Both points satisfy the equation, confirming they lie on the line.

Conclusion

Converting the equation 5x + 3y = 9 to slope-intercept form (y = (-5/3)x + 3) allows us to readily identify its slope (-5/3) and y-intercept (3). This form provides valuable insights into the line's characteristics and facilitates graphical representation and further analysis. Understanding the different forms of linear equations and the ability to convert between them is a fundamental skill in various mathematical applications and problem-solving scenarios. By mastering these concepts, you’ll enhance your ability to analyze and interpret linear relationships effectively.