Problem 3 - Calculus

Question

A line goes through (1,-1) and (4,8) . Write its equation in point-slope form. Then write it as \(y=m x+b\).

Step-by-Step Solution

Answer

The line's equation in point-slope form is \(y + 1 = 3(x - 1)\), and in slope-intercept form, it's \(y = 3x - 4\).

Step 1: Calculate the Slope

To find the slope \(m\) of the line passing through the points \((1, -1)\) and \((4, 8)\), we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1, y_1) = (1, -1)\) and \((x_2, y_2) = (4, 8)\). Substituting the values, we get: \[ m = \frac{8 - (-1)}{4 - 1} = \frac{8 + 1}{3} = \frac{9}{3} = 3. \] So, the slope \(m\) is 3.

Step 2: Write in Point-Slope Form

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). Using the point \((1, -1)\) and the slope \(m = 3\), the equation becomes: \[ y + 1 = 3(x - 1). \] This is the equation of the line in point-slope form.

Step 3: Convert to Slope-Intercept Form

Now, we simplify the equation \( y + 1 = 3(x - 1) \) to the slope-intercept form \( y = mx + b \). Start by distributing the \(3\): \[ y + 1 = 3x - 3. \] Next, subtract \(1\) from both sides to solve for \(y\): \[ y = 3x - 3 - 1. \] Simplify the right-hand side: \[ y = 3x - 4. \] Thus, the equation in slope-intercept form is \(y = 3x - 4\).

Key Concepts

Slope Calculation

Calculating the slope is fundamental in understanding how a line behaves between two points. The slope, often represented as \(m\), is what we call the 'steepness' or 'incline' of a line. It tells us how much the line goes up (or down) for each step we move horizontally. To find the slope between two points like \((1, -1)\) and \((4, 8)\), you can use the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. By substituting the given values, we find:

• \(m = \frac{8 - (-1)}{4 - 1} = \frac{9}{3} = 3\)

The slope is 3, indicating that for every 3 units the line moves vertically, it moves 1 unit horizontally. This calculated slope helps us describe the line more accurately.

Linear Equation

A linear equation represents a straight line on a graph and can be defined in different forms, such as the point-slope form or the slope-intercept form. In our problem, we're initially using the point-slope form, which is:

• \(y - y_1 = m(x - x_1)\)

This form is practical because it quickly shows how the line relates to a specific point, like \((1, -1)\), and the slope \(m = 3\). The equation \(y + 1 = 3(x - 1)\) tells us the relationship between \(x\) and \(y\) given this slope and point.

• By substituting the known values into the point-slope formula, we transform it to a readable form that directly ties the equation to the line's properties.

Starting out with this form can make recognizing important line characteristics more intuitive.

Slope-Intercept Form

The slope-intercept form is perhaps the most common way to express a linear equation. It shows the line's slope and where it crosses the y-axis, making it straightforward for graphing purposes. This form is given by:

• \(y = mx + b\)

Here, \(m\) is the slope and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
To convert from point-slope form \(y + 1 = 3(x - 1)\) to slope-intercept form, we expand and rearrange the terms:

• Start by distributing the \(3\): \(y + 1 = 3x - 3\)
• Subtract \(1\) to isolate \(y\): \(y = 3x - 4\)

The resulting equation \(y = 3x - 4\) shows us that the slope is \(3\) and the y-intercept is \(-4\). This form effectively communicates the position and angle of the line in a direct and visually intuitive way.