What Is The Graph Of 3x 5y 15

What is the Graph of 3x + 5y = 15? A Comprehensive Guide

The equation 3x + 5y = 15 represents a straight line in the Cartesian coordinate system. Understanding its graph involves identifying key features like its intercepts, slope, and how to plot it accurately. This comprehensive guide will delve into these aspects, providing a thorough understanding of the equation's graphical representation.

Understanding Linear Equations

Before diving into the specifics of 3x + 5y = 15, let's refresh our understanding of linear equations. A linear equation is an algebraic equation of the form Ax + By = C, where A, B, and C are constants, and x and y are variables. The graph of a linear equation is always a straight line. This is because every point (x, y) that satisfies the equation lies on the same straight line.

The equation 3x + 5y = 15 perfectly fits this description, with A = 3, B = 5, and C = 15.

Finding the x-intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept, we set y = 0 in the equation and solve for x:

3x + 5(0) = 15 3x = 15 x = 5

Therefore, the x-intercept is (5, 0).

Finding the y-intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept, we set x = 0 in the equation and solve for y:

3(0) + 5y = 15 5y = 15 y = 3

Therefore, the y-intercept is (0, 3).

Calculating the Slope

The slope of a line describes its steepness and direction. It's represented by the letter 'm' and calculated as the change in y divided by the change in x between any two points on the line. The formula for the slope is:

m = (y₂ - y₁) / (x₂ - x₁)

We can use the x-intercept (5, 0) and the y-intercept (0, 3) to calculate the slope:

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

Therefore, the slope of the line is -3/5. A negative slope indicates that the line is decreasing (sloping downwards) from left to right.

Plotting the Graph

Now that we have the x-intercept (5, 0), the y-intercept (0, 3), and the slope (-3/5), we can easily plot the graph.

1. Plot the intercepts: Mark the points (5, 0) and (0, 3) on the Cartesian plane.

2. Use the slope: From the y-intercept (0, 3), move down 3 units (because the slope is negative) and right 5 units. This gives you another point on the line. You can repeat this process to find more points if needed.

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

Alternative Methods for Graphing

While using intercepts and slope is a common and effective method, there are alternative approaches to graphing this linear equation:

1. Solving for y:

We can rewrite the equation in slope-intercept form (y = mx + b), where 'm' is the slope and 'b' is the y-intercept. Solving for y:

3x + 5y = 15 5y = -3x + 15 y = (-3/5)x + 3

This clearly shows the slope (-3/5) and the y-intercept (3). From here, you can plot the y-intercept and use the slope to find other points on the line.

2. Using a Table of Values:

Create a table with x and y values. Choose several x values, substitute them into the equation 3x + 5y = 15, and solve for the corresponding y values. Plot these points and draw a line through them. This method is especially useful when dealing with more complex equations.

3. Using Technology:

Many graphing calculators and online graphing tools can easily plot linear equations. Simply input the equation 3x + 5y = 15 and the software will generate the graph.

Interpreting the Graph

The graph of 3x + 5y = 15 visually represents all the (x, y) pairs that satisfy the equation. Every point on the line is a solution to the equation. The line itself extends infinitely in both directions, indicating that there are infinitely many solutions.

The intercepts provide valuable information: the x-intercept (5, 0) indicates that when y is 0, x is 5. Similarly, the y-intercept (0, 3) shows that when x is 0, y is 3. The slope (-3/5) demonstrates the rate of change of y with respect to x. For every 5-unit increase in x, y decreases by 3 units.

Applications of Linear Equations

Linear equations like 3x + 5y = 15 have numerous applications in various fields, including:

• Economics: Modeling supply and demand, calculating costs and profits.
• Physics: Describing motion, relationships between velocity, acceleration, and time.
• Engineering: Analyzing circuits, designing structures.
• Computer Science: Creating algorithms, modeling data.

Understanding how to graph linear equations is fundamental to solving problems in these and many other areas.

Beyond the Basics: Further Exploration

While this guide provides a comprehensive understanding of graphing 3x + 5y = 15, you can explore further concepts to deepen your knowledge:

• Systems of linear equations: Explore how to solve problems involving multiple linear equations simultaneously.
• Linear inequalities: Learn how to graph inequalities involving linear expressions.
• Linear programming: Apply linear equations and inequalities to optimize solutions in real-world problems.
• Matrices and vectors: Understand how matrices and vectors can be used to represent and manipulate linear equations.

Mastering the ability to graph linear equations is a crucial skill in mathematics and its applications. By understanding the concepts of intercepts, slopes, and various graphing techniques, you can confidently represent and interpret linear relationships. The equation 3x + 5y = 15 serves as a perfect example to solidify this foundational understanding.