Finding X And Y Intercepts For -5x + 7y = 140
Finding the x and y intercepts of a linear equation is a fundamental concept in algebra and is crucial for graphing the line and understanding its behavior. In this comprehensive guide, we will meticulously walk through the process of determining the intercepts for the equation $-5x + 7y = 140$. We will break down each step, providing a clear and concise explanation to ensure you grasp the underlying principles. Understanding intercepts is not only essential for solving equations but also for interpreting real-world scenarios represented by linear functions.
Understanding Intercepts
Before diving into the calculations, it's vital to define what intercepts are. Intercepts are the points where a line crosses the x-axis and the y-axis. The x-intercept is the point where the line intersects the x-axis, and at this point, the y-coordinate is always 0. Conversely, the y-intercept is the point where the line intersects the y-axis, and at this point, the x-coordinate is always 0. Locating these intercepts provides valuable information about the line's position and orientation on the coordinate plane. Knowing the intercepts makes it easier to graph the line and to analyze the relationship between the variables.
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 0. Therefore, we substitute $y = 0$ into the equation and solve for $x$. Let’s apply this to our equation:
Substitute $y = 0$:
Simplify the equation:
Now, divide both sides by -5 to isolate $x$:
Thus, the x-intercept is the point where $x = -28$ and $y = 0$. We can express this as a coordinate pair: $(-28, 0)$. This point is where the line crosses the x-axis. The x-intercept is a crucial point because it tells us where the line intersects the horizontal axis, providing a specific value of x when y is zero. In practical terms, this intercept can represent a starting point, a break-even point, or some other significant value depending on the context of the linear equation.
Finding the y-intercept
To find the y-intercept, we follow a similar process but this time, we determine the point where the line crosses the y-axis. At this point, the x-coordinate is 0. We substitute $x = 0$ into the equation and solve for $y$. Let’s apply this to our equation:
Substitute $x = 0$:
Simplify the equation:
Now, divide both sides by 7 to isolate $y$:
Therefore, the y-intercept is the point where $x = 0$ and $y = 20$. We can express this as a coordinate pair: $(0, 20)$. This point is where the line crosses the y-axis. The y-intercept is equally significant as the x-intercept. It represents the point where the line intersects the vertical axis, giving us the value of y when x is zero. In many real-world applications, the y-intercept represents an initial value or a starting condition. For instance, if the equation models the cost of a service, the y-intercept might represent a fixed initial fee.
Expressing the Intercepts
We have found the x-intercept to be $(-28, 0)$ and the y-intercept to be $(0, 20)$. These are the points where the line $-5x + 7y = 140$ intersects the x and y axes, respectively. Expressing these intercepts as coordinate pairs provides a clear and concise way to represent the points. Each intercept is defined by its x and y coordinates, indicating its exact location on the coordinate plane. Understanding how to represent intercepts in this format is essential for accurately graphing and analyzing linear equations.
Conclusion
In summary, the x-intercept of the equation $-5x + 7y = 140$ is $(-28, 0)$, and the y-intercept is $(0, 20)$. These intercepts are crucial for graphing the line and understanding its behavior. By finding these points, we gain valuable insights into the linear relationship described by the equation. The process of finding intercepts involves substituting 0 for one variable and solving for the other, a fundamental skill in algebra. Mastering this skill allows for a better understanding of linear equations and their applications. Whether you are graphing lines, solving systems of equations, or analyzing real-world problems, the ability to find and interpret intercepts is invaluable.
Understanding how to find x and y intercepts is a critical skill in algebra and coordinate geometry. These intercepts provide key points that help in graphing a linear equation and understanding its behavior. By setting one variable to zero and solving for the other, we can easily identify where the line crosses the axes. In this case, the x-intercept is found by setting $y$ to zero, and the y-intercept is found by setting $x$ to zero. This method is applicable to any linear equation and provides a straightforward way to visualize and analyze the line's position in the coordinate plane. Intercepts are not just mathematical concepts; they often have practical interpretations in real-world applications, such as representing starting points, break-even points, or initial conditions in various scenarios.
Therefore:
x$ intercept = $(-28, 0)
y$ intercept = $(0, 20)
