Find X And Y Intercepts Of -3x-5y=-30
Finding the x and y intercepts of a linear equation is a fundamental concept in algebra. These intercepts represent the points where the line crosses the x-axis and the y-axis, respectively. Determining these points helps us to visualize and understand the behavior of the line, making it easier to graph and analyze. In this article, we will walk through the process of finding the x and y intercepts of the equation -3x - 5y = -30, providing a clear and concise explanation for each step.
Understanding Intercepts
Before diving into the calculations, let's briefly define what x and y intercepts are. The x-intercept is the point where the line intersects the x-axis. At this point, the y-coordinate is always 0. Conversely, the y-intercept is the point where the line intersects the y-axis, and the x-coordinate is always 0. Understanding this basic principle is crucial for solving linear equations and graphing lines accurately.
Finding the x-intercept
To find the x-intercept, we need to determine the point where the line crosses the x-axis. As mentioned earlier, at this point, the y-coordinate is always 0. Therefore, we substitute y = 0 into the equation and solve for x. This process effectively isolates the x value, giving us the x-coordinate of the intercept. Substituting y = 0 into the equation -3x - 5y = -30 yields -3x - 5(0) = -30. Simplifying this equation, we get -3x = -30. To solve for x, we divide both sides of the equation by -3, resulting in x = 10. Thus, the x-intercept is the point (10, 0). This means the line crosses the x-axis at the point where x is 10 and y is 0. This is a crucial piece of information for graphing the line, as it provides one of the two points needed to draw a straight line. Understanding how to find the x-intercept is essential for anyone studying linear equations, as it helps in visualizing the equation's graph and understanding its behavior. In addition to graphing, finding the x-intercept is useful in various real-world applications, such as determining break-even points in business or analyzing physical phenomena described by linear relationships. The process of setting y = 0 and solving for x is a fundamental technique in algebra and is widely applicable across different mathematical contexts. Mastering this skill is a significant step in understanding linear functions and their graphical representations.
Finding the y-intercept
Next, to find the y-intercept, we need to determine the point where the line crosses the y-axis. As we know, at this point, the x-coordinate is always 0. Therefore, we substitute x = 0 into the equation and solve for y. This process isolates the y value, providing us with the y-coordinate of the intercept. Substituting x = 0 into the equation -3x - 5y = -30, we get -3(0) - 5y = -30. Simplifying this equation, we have -5y = -30. To solve for y, we divide both sides of the equation by -5, which gives us y = 6. Therefore, the y-intercept is the point (0, 6). This means the line crosses the y-axis at the point where x is 0 and y is 6. This point is another crucial piece of information for graphing the line. The y-intercept, along with the x-intercept, provides two distinct points that uniquely define the line in a two-dimensional plane. Understanding how to calculate the y-intercept is just as important as finding the x-intercept, as both points together give a comprehensive view of the line's position and orientation. In practical applications, the y-intercept can represent an initial value or starting point in a given scenario, such as the initial cost of a project or the starting amount of a substance in a chemical reaction. The ability to find the y-intercept is thus a valuable skill not only in mathematics but also in various fields that use linear models. The step-by-step process of setting x to 0 and solving for y is a straightforward yet powerful technique that is essential for analyzing linear equations and their real-world implications.
Solution
In summary, we have found the x and y intercepts of the equation -3x - 5y = -30. The x-intercept is the point (10, 0), which means the line crosses the x-axis at x = 10. The y-intercept is the point (0, 6), indicating that the line crosses the y-axis at y = 6. These two points are essential for graphing the line accurately and understanding its behavior in the coordinate plane. The process of finding these intercepts involves substituting 0 for one variable and solving for the other, a fundamental technique in algebra. By understanding how to find these intercepts, you can quickly visualize the position and orientation of the line, which is crucial for solving various mathematical problems and real-world applications. The x and y intercepts provide a clear picture of how the line interacts with the coordinate axes, making it easier to analyze and interpret the equation. This skill is not only valuable in mathematics but also in fields that rely on linear models, such as economics, physics, and engineering. The ability to find intercepts allows for a more intuitive understanding of linear relationships and their implications. Therefore, mastering this technique is a key step in developing a strong foundation in algebra and its practical applications. Additionally, understanding intercepts is crucial for more advanced topics in mathematics, such as solving systems of equations and understanding the behavior of functions. The intercepts provide key reference points that aid in visualizing and analyzing mathematical relationships, making them an indispensable tool in the study of mathematics. Therefore, the process of finding x and y intercepts is a fundamental skill that is essential for anyone looking to deepen their understanding of linear equations and their applications.
**x** intercept = (10 ,0)
**y** intercept = (0, 6)
