Problem 42 - 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 (3,-1) and (6,0)

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{3}x - 2 \).

Step 1: Identify the points

The line passes through the points (3, -1) and (6, 0). These two points will be used to find the slope and thus the equation of the line.

Step 2: Calculate the slope

Use the formula for slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \) where \((x_1, y_1)\) is (3, -1) and \((x_2, y_2)\) is (6, 0). Substitute the coordinates into the formula.\[ m = \frac{0 - (-1)}{6 - 3} = \frac{1}{3} \]

Step 3: Use point-slope form

With the slope \(m = \frac{1}{3}\) and using one point, (3, -1), substitute into the point-slope form of a line, \( y - y_1 = m(x - x_1) \):\[ y - (-1) = \frac{1}{3}(x - 3) \]

Step 4: Simplify to slope-intercept form

Distribute and simplify to get the equation into \(y = mx + b\) form:\[ y + 1 = \frac{1}{3}x - 1 \]Subtract 1 from both sides:\[ y = \frac{1}{3}x - 2 \]

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. It's often represented by the letter \( m \). To find the slope between two points, use the formula:

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

This formula calculates the change in the \( y \)-coordinates divided by the change in the \( x \)-coordinates. In simpler terms, it tells you how much the line goes up (or down) for each step it goes across.For our example, using the points \((3, -1)\) and \((6, 0)\), we substitute:

• The difference in \( y \)-coordinates: \( 0 - (-1) = 1 \)
• The difference in \( x \)-coordinates: \( 6 - 3 = 3 \)

So, the slope \( m = \frac{1}{3} \). This means the line rises 1 unit for every 3 units it moves to the right.

Slope-Intercept Form

The slope-intercept form of a line is expressed as \( y = mx + b \). This equation makes it easy to identify the slope and the \( y \)-intercept:

• \( m \) is the slope of the line.
• \( b \) is the \( y \)-intercept, where the line crosses the \( y \)-axis.

In our solution, we simplified the equation to this form. From the point-slope form, we rearranged it and found:

• \( y = \frac{1}{3}x - 2 \)

Here, the slope \( m \) is \( \frac{1}{3} \), and the \( y \)-intercept \( b \) is \(-2\). This form is particularly useful because it provides a clear picture of how the line behaves just by looking at the equation.

Point-Slope Form

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), and is very useful when you know a point on the line and the slope. Here's how it works:

• \( (x_1, y_1) \) is a point on the line.
• \( m \) is the slope.

In our example, after calculating the slope \( m = \frac{1}{3} \), we used the point \( (3, -1) \). Substituting into the formula gives:

• \( y + 1 = \frac{1}{3}(x - 3) \)

This form quickly converts to other forms like the slope-intercept form. It is particularly handy because it's straightforward to use when you have a specific point and the slope of the line.