Problem 41 - Brief Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Passing through the points \((5,3)\) and \((7,-1)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -2x + 13\).

Step 1: Identify Known Coordinates

We are given two points through which the line passes: \((x_1, y_1) = (5, 3)\) and \((x_2, y_2) = (7, -1)\). These points will be used to determine the slope and equation of the line.

Step 2: Calculate the Slope (m)

The slope \(m\) of the line can be calculated using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]\[m = \frac{-1 - 3}{7 - 5} = \frac{-4}{2} = -2\]So, the slope of the line is \(m = -2\).

Step 3: Use Point-Slope Form

With the slope \(m = -2\) and using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), choose the point \((5, 3)\) to plug in: \[y - 3 = -2(x - 5)\]

Step 4: Simplify to Slope-Intercept Form

Distribute and rearrange the equation to get it in the form \(y = mx + b\): \[y - 3 = -2x + 10\]\[y = -2x + 10 + 3\] \[y = -2x + 13\]This is the equation of the line in slope-intercept form.

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness and direction. To calculate the slope, you can use two points that the line passes through. In this example, we have the points \((5,3)\) and \((7,-1)\). The slope formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \( m \) represents the slope.Plug the coordinates into the formula:

• \( m = \frac{-1 - 3}{7 - 5} = \frac{-4}{2} = -2 \)

The slope \( m \) is \(-2\). This tells us the line decreases as you move from left to right, and for every unit you move horizontally, it moves down 2 units vertically. Understanding slope is crucial because it describes how a line moves and its rate of change. Calculating the slope is the first critical step in finding an equation of a line.

Point-Slope Form

The point-slope form is a way to express the equation of a line using the slope calculated and a point that the line passes through. It's handy when you already know the slope and a specific point on the line.The general formula of point-slope form is:

• \( y - y_1 = m(x - x_1) \)

In this situation, we use the slope \(-2\) and the point \((5, 3)\). Let's substitute them into the formula:

• \( y - 3 = -2(x - 5) \)

This creates a more initial form of the line equation starting from a known point with a known slope. The point-slope form is especially beneficial when you’re focusing on a segment of a line between two points or when multiple calculations involve different points. It's a stepping stone to rearranging into the more common slope-intercept format.

Slope-Intercept Form

The slope-intercept form is the most common way to express the equation of a line. This form includes the slope and the y-intercept, making it straightforward to graph and understand the relationship between the variables.The formula for the slope-intercept form is:

• \( y = mx + b \)

Where \( m \) denotes the slope, and \( b \) represents the y-intercept.From point-slope form \( y - 3 = -2(x - 5) \), our task is to simplify this into slope-intercept form. By distributing and rearranging, we find:

• \( y - 3 = -2x + 10 \)
• \( y = -2x + 10 + 3 \)
• \( y = -2x + 13 \)

Thus, the slope-intercept form of the line is \( y = -2x + 13 \). Here, \(-2\) is the slope, indicating how steep the line is, and \(13\) is the y-intercept, which tells where the line crosses the y-axis. Having this form makes it easy to plot the line and understand its behavior.