Problem 51 - Calculus with Concepts in Calculus

Question

Find a two-point equation of the given line. The line containing \((-2,4)\) and \((-1,3)\)

Step-by-Step Solution

Answer

The two-point equation of the line is \(y = x + 6\).

Step 1: Understand the Two-Point Form

The two-point form of a line's equation is derived using two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line. The equation is given as \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \). Here, \((x_1, y_1) = (-2, 4)\) and \((x_2, y_2) = (-1, 3)\).

Step 2: Calculate the Slope

First, calculate the slope \(m\) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points, we get:\[m = \frac{3 - 4}{-1 - (-2)} = \frac{-1}{-1} = 1\]So, the slope \(m\) is 1.

Step 3: Formulate the Equation

Using one of the points, say \((-2, 4)\), and the slope \(m = 1\), write the equation in point-slope form:\[y - 4 = 1(x + 2)\]This is the equation of the line in point-slope form.

Step 4: Simplify the Equation

Expand and simplify the equation obtained:\[y - 4 = 1(x + 2)\]\[y - 4 = x + 2\]Add 4 to both sides to isolate \(y\):\[y = x + 6\]This is the simplified equation of the line.

Key Concepts

Understanding Slope Calculation

Slope is a key concept in mathematics that describes how steep a line is. It's like measuring how fast you go up or down as you move along the line. When you have two points on a line, calculating the slope helps you understand the line's direction.
To find the slope, you need to look at how changes occur between the points. For two points \( (x_1, y_1) \) and \( (x_2, y_2) \), you can find the slope \( m \) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• First, subtract the \( y \)-coordinates: \( y_2 - y_1 \).
• Then, subtract the \( x \)-coordinates: \( x_2 - x_1 \).
• Divide the change in \( y \) by the change in \( x \).

This gives you the slope.
In our exercise, using the points \( (-2, 4) \) and \( (-1, 3) \), we calculated the slope as 1. This tells us that for every step you take sideways, you move up by the same amount along the line.

Exploring the Point-Slope Form

Once you have the slope, you can use it to write the equation of the line. One of the most helpful forms for doing this is the point-slope form. This form helps us express the line with the information we already know: a point on the line and the slope. It's like having a recipe for the line.
The point-slope form is written as: \[ y - y_1 = m(x - x_1) \]

• \( y_1 \) and \( x_1 \) are the coordinates of any point on the line.
• \( m \) is the slope we calculated.

From our example, with a slope of 1 and using the point \( (-2, 4) \), the equation becomes: \[ y - 4 = 1(x + 2) \]
This equation is like a building block, which you can further expand or change into other forms.

Creating a Line Equation

The final step in understanding line equations is turning a point-slope form into an easily recognizable line equation. Often, the goal is to find a simple relationship between \( x \) and \( y \).
To do this, simplify the point-slope form. In the exercise, we simplified \[ y - 4 = 1(x + 2) \] by distributing the slope (1) and then re-arranging it:

• Expand: \( y - 4 = x + 2 \).
• Add 4 to both sides to solve for \( y \): \( y = x + 6 \).

Now we have a clear equation, \( y = x + 6 \), which is in the slope-intercept form.
This form is familiar because it directly shows the slope and \( y \)-intercept, making it easy to graph or use in further calculations.