Problem 5 - Applied Calculus

Question

Find an equation for the line that passes through the given points. $$ (0,2) \text { and }(2,3) $$

Step-by-Step Solution

Answer

The line equation is \(y = \frac{1}{2}x + 2\).

Step 1: Identify the points

We begin by identifying the given points: \((0,2)\) and \((2,3)\). The line passes through these two points.

Step 2: Calculate the slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points, \(m = \frac{3 - 2}{2 - 0} = \frac{1}{2}\). Thus, the slope of the line is \(\frac{1}{2}\).

Step 3: Use the point-slope form of a line equation

The point-slope form of the line equation is \(y - y_1 = m(x - x_1)\). We can use point \((0, 2)\) for \((x_1, y_1)\). Substitute the values into the equation: \(y - 2 = \frac{1}{2}(x - 0)\).

Step 4: Simplify the equation

Simplify the equation \(y - 2 = \frac{1}{2}x\) by adding 2 to both sides: \(y = \frac{1}{2}x + 2\). This is the equation of the line in slope-intercept form \(y = mx + c\).

Step 5: Verify the equation with both points

To ensure the equation is correct, substitute \(x = 0\), it gives \(y = 2\), and substitute \(x = 2\), it gives \(y = 3\). Both points satisfy the equation, confirming our solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly used forms when dealing with linear relationships. This form is particularly handy because it clearly shows both the slope and the y-intercept of the line in a straightforward way. The general equation for the slope-intercept form is \( y = mx + c \). Here, \( m \) represents the slope of the line, and \( c \) is the y-intercept, which is the point where the line crosses the y-axis.

To understand this more deeply, let's see what each part of the equation signifies:

• \( m \): This number tells us how steep the line is and in which direction it slants. A positive slope means the line goes upwards, while a negative slope indicates it goes downwards as you move along the x-axis.
• \( c \): This value is where the line crosses the y-axis. It tells us what the value of \( y \) is when \( x = 0 \).

In the example given, the equation of the line is \( y = \frac{1}{2}x + 2 \). Here, the slope \( \frac{1}{2} \) shows how y increases by \( \frac{1}{2} \) for each increase of 1 in x. Meanwhile, the \( c \) value, which is 2, reveals the line passes through the point \( (0, 2) \) on the y-axis.

Point-Slope Form

The point-slope form is another useful way to express the equation of a line, especially when you know a point on the line and the slope. This form is represented by the equation \( y - y_1 = m(x - x_1) \).

Here's what the terms mean:

• \( (x_1, y_1) \): These are the coordinates of a specific point on the line.
• \( m \): This is the slope of the line, similar to what we have in the slope-intercept form.

This form is very interactive because you can easily plug in any given point \((x_1, y_1)\) and a slope \(m\) to get the linear equation. In the example, point \((0, 2)\) was used with a slope of \(\frac{1}{2}\). By substituting these into the formula, the equation becomes:\[ y - 2 = \frac{1}{2}(x - 0) \]Once written in this form, if needed, you can further simplify it into the slope-intercept form to reveal more characteristics of the line.

Calculating Slope

Calculating the slope is a fundamental step when working with linear equations. The slope measures the steepness and direction of a line and can be calculated using two known points on the line.

The formula to find the slope \( m \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula represents the "rise over run," signifying the change in y over the change in x.

• \( y_2 - y_1 \): This is the "rise," or the difference in the y-coordinates between the two points.
• \( x_2 - x_1 \): This is the "run," or the difference in the x-coordinates between the two points.

Using point (0, 2) and (2, 3), we substituted into the slope formula:\[ m = \frac{3 - 2}{2 - 0} = \frac{1}{2} \]The slope here is \(\frac{1}{2}\), indicating that for every increase of 2 in x, y increases by 1. Understanding how to calculate the slope allows one to discern a lot about the nature of the line and is crucial for forming the equations of lines in both slope-intercept and point-slope forms.