Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((3,-8)\) and \((5,-3)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{5}{2}x - \frac{31}{2} \).

Step 1: Identify the Coordinates

We are given two points on the line: \((3, -8)\) and \((5, -3)\). Identify these as \((x_1, y_1)\) and \((x_2, y_2)\). Here, \((x_1, y_1) = (3, -8)\) and \((x_2, y_2) = (5, -3)\).

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 calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points: \[ m = \frac{-3 - (-8)}{5 - 3} = \frac{5}{2} \] Thus, the slope of the line is \( \frac{5}{2} \).

Step 3: Write the Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept. We have already found \(m = \frac{5}{2}\).

Step 4: Find the Y-Intercept

To find the y-intercept \(b\), use the slope-intercept form with one of the given points, say \((3, -8)\).Substitute into \(y = mx + b\): \[ -8 = \frac{5}{2}(3) + b \]This simplifies to \[ -8 = \frac{15}{2} + b \]Subtract \(\frac{15}{2}\) from \(-8\): \[ b = -8 - \frac{15}{2} = -\frac{31}{2} \] Thus, \(b\) is \(-\frac{31}{2}\).

Step 5: Write Final Equation

Now substitute the slope \(m = \frac{5}{2}\) and y-intercept \(b = -\frac{31}{2}\) back into the slope-intercept form: \( y = \frac{5}{2}x - \frac{31}{2} \). This is the equation of the line in slope-intercept form.

Key Concepts

Equation of a Line

An equation of a line can be represented in different forms, but one of the most common and useful forms is the slope-intercept form. This form allows you to easily identify and understand the behavior of a linear relationship. In this exercise, we are focusing on finding the slope-intercept form of a line using two given points. The slope-intercept form is expressed as:

• \( y = mx + b \)

where \( m \) is the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
This form is particularly useful because it provides a direct way to see how changes in \( x \) values affect changes in \( y \) values using the slope \( m \). Additionally, knowing the y-intercept \( b \) gives you a clear understanding of the real world context or initial value when \( x = 0 \).

The step-by-step solution transforms the points into an equation, ensuring the line is defined in terms of its slope and y-intercept.

Slope Calculation

One of the first things you need to do when finding the equation of a line is to calculate its slope. The slope is a measure of how steep a line is and is calculated based on the change in \( y \)-coordinates over the change in \( x \)-coordinates between two points on the line.

• The formula to calculate the slope \( m \) is: \ \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula tells us how much \( y \) changes for a certain change in \( x \).

In our example:

• For the points \((3, -8)\) and \((5, -3)\), substitute into the formula: \ \[ m = \frac{-3 - (-8)}{5 - 3} = \frac{5}{2} \]

The slope \( \frac{5}{2} \) means for every 2 units increase in \( x \), \( y \) increases by 5 units.
Understanding slope is fundamental in coordinate geometry as it describes a line's inclination and direction.

Coordinate Geometry

Coordinate Geometry provides a robust framework to explore geometric concepts using algebraic principles. By plotting points on a graph, you can derive information about lines and shapes. In coordinate geometry, each point has an \( (x, y) \) representation, where \( x \) is the horizontal coordinate and \( y \) is the vertical coordinate.

When you're given two points like \((3, -8)\) and \((5, -3)\), they define a unique line on the plane. Using these coordinates, you can determine significant properties of the line, such as its slope, as we've done. Understanding how lines intersect and how their slopes interact gives insight into their relative positions and can help solve real-world problems.

This concept allows us to graphically represent mathematical expressions and physically visualize their properties. This visualization helps simplify complex ideas and solve equations graphically, where numerical calculations are involved.