Finding The Ratio Of Intercepts For 3x - 4y = 17

In the realm of coordinate geometry, linear equations hold a fundamental place. They describe straight lines, the simplest of curves, and provide a foundation for understanding more complex geometric concepts. Among the various forms of representing a line, the standard form ($Ax + By = C$) and the intercept form are particularly insightful. This article delves into the equation $3x - 4y = 17$, exploring its intercepts and ultimately determining the ratio ${ \frac{c}{k} }$, where ${ c }$ is the x-intercept and ${ k }$ is the y-intercept. This exploration will involve finding the points where the line intersects the x and y axes and then calculating the ratio. Understanding intercepts is crucial in various fields, including mathematics, physics, and economics, as they represent key values where a function or relationship crosses the axes, providing essential information about the scenario being modeled.

Before we dive into the specifics of the equation $3x - 4y = 17$, let's establish a solid understanding of linear equations and intercepts. A linear equation in two variables (typically x and y) can be written in the general form $Ax + By = C$, where A, B, and C are constants. The graph of such an equation is a straight line. The intercepts of this line are the points where it crosses the x-axis and 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. Similarly, the y-intercept is the point where the line intersects the y-axis, and at this point, the x-coordinate is always 0. Intercepts are valuable because they give us two specific points on the line, which can be used to graph the line or to understand the relationship between the variables. Finding intercepts is a fundamental skill in algebra and coordinate geometry, with wide-ranging applications in various mathematical and real-world contexts. Understanding linear equations and how to determine their intercepts is crucial for solving problems involving graphs and equations. These intercepts give us key information about the line's position and orientation in the coordinate plane.

To find the x-intercept of the line represented by the equation $3x - 4y = 17$, we set $y = 0$ in the equation. This is because the x-intercept is the point where the line crosses the x-axis, and on the x-axis, the y-coordinate is always zero. Substituting $y = 0$ into the equation, we get:

$3x - 4(0) = 17$

Simplifying the equation, we have:

$3x = 17$

Now, we solve for $x$ by dividing both sides of the equation by 3:

$x = \frac{17}{3}$

Therefore, the x-intercept is the point $\left(\frac{17}{3}, 0\right)$. This means that the line crosses the x-axis at the point where x is equal to ${ \frac{17}{3} }$ and y is 0. The x-intercept is a crucial point on the graph of the line, as it represents the value of x when y is zero. It is often used in applications to find the starting point or the initial value of a linear function. Finding the x-intercept involves setting the y-coordinate to zero and solving for x. This process helps us locate the point where the line intersects the x-axis, which is a critical piece of information for graphing and analyzing the line. Understanding how to calculate the x-intercept is essential for working with linear equations and their graphical representations.

The y-intercept is the point where the line intersects the y-axis. At this point, the x-coordinate is always 0. To find the y-intercept of the line $3x - 4y = 17$, we set $x = 0$ in the equation. This substitution gives us:

$3(0) - 4y = 17$

Simplifying the equation, we get:

$-4y = 17$

To solve for $y$, we divide both sides of the equation by -4:

$y = -\frac{17}{4}$

Thus, the y-intercept is the point $\left(0, -\frac{17}{4}\right)$. This indicates that the line crosses the y-axis at the point where y is equal to ${ -\frac{17}{4} }$ and x is 0. The y-intercept is another key point on the line, representing the value of y when x is zero. In practical applications, the y-intercept can represent the starting value or initial condition of a linear relationship. Finding the y-intercept is a similar process to finding the x-intercept, but here we set the x-coordinate to zero and solve for y. This gives us the point where the line intersects the y-axis, which is vital for understanding the line's position and behavior. The ability to calculate the y-intercept is a fundamental skill in working with linear equations and their graphs.

Now that we have found the x-intercept and the y-intercept, we can calculate the ratio ${ \frac{c}{k} }$, where ${ c }$ is the x-intercept and ${ k }$ is the y-intercept. We found that the x-intercept is ${ c = \frac{17}{3} }$ and the y-intercept is ${ k = -\frac{17}{4} }$. So, we need to find the value of:

$\frac{c}{k} = \frac{\frac{17}{3}}{-\frac{17}{4}}$

To divide fractions, we multiply by the reciprocal of the denominator:

$\frac{c}{k} = \frac{17}{3} \times \left(-\frac{4}{17}\right)$

We can simplify this expression by canceling out the common factor of 17:

$\frac{c}{k} = \frac{1}{3} \times (-4)$

Multiplying the fractions, we get:

$\frac{c}{k} = -\frac{4}{3}$

Therefore, the value of the ratio ${ \frac{c}{k} }$ is ${ -\frac{4}{3} }$. This ratio represents the relationship between the x and y intercepts of the line. Understanding this ratio can be helpful in various applications, such as comparing the intercepts or analyzing the line's slope. Calculating the ratio of intercepts involves using the values of the x and y intercepts that we have previously found. This ratio provides a concise way to describe the relationship between the intercepts and is a valuable tool for analyzing linear equations and their graphs. The final result of -\frac{4}{3} is a key characteristic of the line $3x - 4y = 17$, giving us insight into its behavior in the coordinate plane.

In this article, we have explored the linear equation $3x - 4y = 17$ and determined the ratio ${ \frac{c}{k} }$, where ${ c }$ is the x-intercept and ${ k }$ is the y-intercept. We found the x-intercept to be ${ \frac{17}{3} }$ and the y-intercept to be ${ -\frac{17}{4} }$. By calculating the ratio, we found that ${ \frac{c}{k} = -\frac{4}{3} }$. This process involved understanding the fundamental concepts of linear equations, intercepts, and how to find them. Intercepts are crucial points on a line that provide valuable information about its position and orientation in the coordinate plane. The ratio of intercepts gives us a concise way to describe the relationship between these points. Understanding linear equations and their properties is essential in mathematics and various fields that utilize mathematical modeling. The ability to find intercepts and calculate their ratios is a valuable skill that allows us to analyze and interpret linear relationships effectively. This exploration not only reinforces these fundamental concepts but also highlights their practical significance in problem-solving and real-world applications. Through this analysis, we've gained a deeper understanding of the line $3x - 4y = 17$ and its characteristics within the coordinate system.