Problem 61 - Contemporary Precalculus

Question

Find an equation for the line satisfying the given conditions. \(x\) -intercept 5 and \(y\) -intercept -5.

Step-by-Step Solution

Answer

Short Answer: The equation of the line with an x-intercept of 5 and a y-intercept of -5 is \(y = x - 5\).

Step 1: Find the slope of the line

The x-intercept is the point where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis. Since we are given the x-intercept (5) and y-intercept (-5), we can write the intercepts as points: \((5,0)\) and \((0,-5)\). To find the slope of the line, we'll use the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. Plugging in the intercepts, we get: \(m = \frac{-5 - 0}{0 - 5} = \frac{-5}{-5} = 1\) Thus, the slope of the line is 1.

Step 2: Use the slope-intercept form to find the equation of the line

Now that we have the slope, we can use the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Since we already know the y-intercept (-5), we can plug in the values: \(y = 1x - 5\)

Step 3: Write the final equation of the line

So the equation of the line that satisfies the given conditions is: \(y = x - 5\)

Key Concepts

X-Intercept

When we discuss the x-intercept of a line, we're referring to the specific point at which the line crosses the x-axis. In simpler terms, it's the location on the graph where the line hits the x-axis and the value of y is zero.

In the context of our problem, the x-intercept is given as 5, which means the line passes through the point \(5, 0\). This point is crucial because, along with the y-intercept, it allows us to calculate the slope of the line, which is a measure of how 'steep' the line is. This is part of what makes our line unique - there's only one line that can pass through \(5, 0\) with a specific slope, and our job is to find its equation.

Y-Intercept

Contrary to the x-intercept, the y-intercept is where the line crosses the y-axis. To visualize it, picture the vertical y-axis and the point where the line touches it—that's the y-intercept, where the x-value is zero.

In the given exercise, the line's y-intercept is at -5, represented as the point \(0, -5\). Knowing this point provides us with half of what we need to construct the equation of the line. It's also directly used when we write the equation in slope-intercept form because the y-intercept is the 'b' in the equation \(y = mx + b\).

Slope of a Line

The slope of a line is a critical concept in algebra and geometry that describes the direction and steepness of the line. It's often denoted as 'm' and calculated by determining the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Mathematically, we express it as \(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\). In our example, the slope is determined using the x-intercept and y-intercept. The calculation \(\frac{-5 - 0}{0 - 5}\) shows how we use the coordinates of the two intercepts to find that the slope 'm' is equal to 1. This defines how the line 'tilts' on the graph. With a slope of 1, we know our line increases at a 45-degree angle, indicating that for every one unit the line moves right, it also moves one unit up.

Slope-Intercept Form

The slope-intercept form is an elegant way to represent the equation of a straight line. It's given as \(y = mx + b\), where 'm' stands for the slope of the line, and 'b' signifies the y-intercept. This form is incredibly user-friendly because it directly shows both the slope and the y-intercept, making it easy to graph a line or understand its characteristics at a glance.

In the solution of our problem, we applied the slope-intercept form by plugging in the slope \(m = 1\) and our y-intercept \(b = -5\), resulting in the equation \(y = x - 5\). This tells us that for this particular line, as x increases by any amount, y increases by the same amount due to the slope of 1, and it will cross the y-axis at -5.