Problem 8 - Applied Calculus

Question

Find an equation for the line that passes through the given points. $$ (4,5) \text { and }(2,-1) $$

Step-by-Step Solution

Answer

The equation is \(y = 3x - 7\).

Step 1: Identify Points

Our points are \((x_1, y_1) = (4, 5)\) and \((x_2, y_2) = (2, -1)\).

Step 2: Calculate Slope

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points, we have \(m = \frac{-1 - 5}{2 - 4} = \frac{-6}{-2} = 3\).

Step 3: Use Point-Slope Form

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\). Using point \((4,5)\) and slope \(m = 3\), the equation becomes \(y - 5 = 3(x - 4)\).

Step 4: Simplify to Slope-Intercept Form

Simplify the equation \(y - 5 = 3(x - 4)\) to get \(y = 3x - 12 + 5\), which simplifies to \(y = 3x - 7\).

Key Concepts

Point-Slope Form

The point-slope form is a way to express the equation of a straight line. This form is particularly helpful when we know a point on the line and the slope. The format of the equation is given by:

• \( y - y_1 = m(x - x_1) \)

In this equation:

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

To use this form, you just need to substitute in values for \((x_1, y_1)\) and \(m\). For example, if the point is \((4, 5)\) and the slope \(m\) is discovered to be 3, the equation becomes \(y - 5 = 3(x - 4)\). Using this form helps to directly insert the slope and point into the equation, efficiently sliding into the slope-intercept form in later steps.

Slope-Intercept Form

The slope-intercept form of the equation of a line is one of the most common ways to write the equation, due to its simplicity and ease of use with graphs. This form provides a quick view of the line's slope and where it crosses the y-axis.
It is written as:

• \( y = mx + b \)

In this arrangement:

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

From a point-slope form like \(y - 5 = 3(x - 4)\), you can easily alter it into the slope-intercept form by expanding and simplifying. Distribute the 3 into the terms \((x - 4)\) to get \(y - 5 = 3x - 12\). Then, add 5 to both sides to obtain \(y = 3x - 7\). This is now in slope-intercept form and clearly shows the slope, 3, and the y-intercept, -7.

Calculating Slope

Calculating the slope of a line is a fundamental step when working with linear equations. The slope, often represented as \(m\), tells you how steep the line is and the direction it's going.
The formula for calculating slope using two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

This equation defines slope as "rise over run," or how much the line moves up or down for every step it takes left or right. A positive slope means the line goes upwards as it moves right, while a negative slope means it goes downwards.
For example, using points \( (4, 5) \) and \( (2, -1) \), substitute into the formula: \( m = \frac{-1 - 5}{2 - 4} = \frac{-6}{-2} = 3 \). So, the line has a slope of 3, indicating it rises by 3 units for every 1 unit it runs to the right.