Finding X And Y Intercepts Of -2x - 5y = 20 A Step-by-Step Guide
In mathematics, particularly in coordinate geometry, intercepts play a crucial role in understanding the behavior of linear equations and their graphical representations. The x and y intercepts are the points where a line crosses the x-axis and y-axis, respectively. These points provide valuable information about the equation and its graph, making it easier to visualize and analyze the relationship between the variables. In this article, we will delve into the process of finding the x and y intercepts of the linear equation -2x - 5y = 20. We will explore the underlying concepts, provide a step-by-step solution, and discuss the significance of intercepts in various mathematical contexts.
Before we dive into the solution, let's first establish a clear understanding of what intercepts are and why they are important. The x-intercept is the point where the line intersects the x-axis. At this point, the y-coordinate is always zero. Similarly, the y-intercept is the point where the line intersects the y-axis, and at this point, the x-coordinate is always zero. These intercepts are essentially the points where one variable is zero, allowing us to isolate and solve for the other variable. Finding the intercepts is a fundamental skill in algebra and is often used in graphing linear equations, solving systems of equations, and analyzing real-world problems that can be modeled linearly. Understanding intercepts helps in visualizing the line's position and orientation in the coordinate plane, which is crucial in many applications of linear equations.
Now, let's proceed with finding the x and y intercepts of the equation -2x - 5y = 20. We will follow a systematic approach to ensure accuracy and clarity.
Finding the X-intercept
To find the x-intercept, we set y = 0 in the equation and solve for x. This is because, at the x-intercept, the y-coordinate is always zero. The equation becomes:
-2x - 5(0) = 20
Simplifying the equation:
-2x = 20
Now, we divide both sides by -2 to isolate x:
x = 20 / -2
x = -10
Therefore, the x-intercept is the point (-10, 0).
Finding the Y-intercept
To find the y-intercept, we set x = 0 in the equation and solve for y. This is because, at the y-intercept, the x-coordinate is always zero. The equation becomes:
-2(0) - 5y = 20
Simplifying the equation:
-5y = 20
Now, we divide both sides by -5 to isolate y:
y = 20 / -5
y = -4
Therefore, the y-intercept is the point (0, -4).
Summary of Intercepts
- X-intercept: (-10, 0)
- Y-intercept: (0, -4)
To further illustrate the significance of these intercepts, let's consider the graphical representation of the equation -2x - 5y = 20. The x-intercept (-10, 0) is the point where the line crosses the x-axis, and the y-intercept (0, -4) is the point where the line crosses the y-axis. By plotting these two points on a coordinate plane, we can easily draw the line. The line will pass through both intercepts, providing a visual representation of the equation. The x-intercept indicates the value of x when y is zero, and the y-intercept indicates the value of y when x is zero. These points serve as anchors for the line, making it easier to sketch the graph accurately. The graphical representation not only confirms our calculations but also provides a visual understanding of the linear relationship between x and y.
Intercepts are not just theoretical points; they have practical applications in various fields. In real-world scenarios, intercepts can represent meaningful values. For instance, in a linear equation representing the cost of a service over time, the y-intercept might represent the initial cost, and the x-intercept might represent the time at which the service is fully paid off. In business, the x-intercept of a cost function could represent the break-even point, where the total cost equals total revenue. Similarly, in physics, intercepts can represent initial conditions or final states in a linear model. Understanding intercepts helps in interpreting the context of the problem and extracting relevant information. In fields like economics, engineering, and even social sciences, linear models are frequently used, and the ability to find and interpret intercepts is a valuable skill.
In conclusion, finding the x and y intercepts of the equation -2x - 5y = 20 is a straightforward process that involves setting one variable to zero and solving for the other. The x-intercept is (-10, 0), and the y-intercept is (0, -4). These intercepts are crucial points that help us understand the behavior and graphical representation of the linear equation. Intercepts have practical applications in various fields, making them an essential concept in mathematics and its applications. By mastering the technique of finding intercepts, you gain a powerful tool for analyzing linear relationships and solving real-world problems. The ability to interpret and use intercepts effectively enhances your problem-solving skills and provides a deeper understanding of linear equations and their significance.
The intercepts are:
- x intercept = (-10, 0)
- y intercept = (0, -4)
