Problem 16 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((3,6)\) and \((0,10)\)

Step-by-Step Solution

Answer

The slope-intercept form is \( y = -\frac{4}{3}x + 10 \).

Step 1: Identify the formula for slope

To find the slope-intercept form, we start by calculating the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) = (3, 6) \) and \( (x_2, y_2) = (0, 10) \).

Step 2: Calculate the slope

Substitute the values into the slope formula: \( m = \frac{10 - 6}{0 - 3} = \frac{4}{-3} = -\frac{4}{3} \). This is the slope of the line.

Step 3: Use point-slope to find equation

Using the point-slope form, \( y - y_1 = m(x - x_1) \), choose the point \((3,6)\). Substitute \( m = -\frac{4}{3} \), \( x_1 = 3 \), and \( y_1 = 6 \) into the equation: \( y - 6 = -\frac{4}{3}(x - 3) \).

Step 4: Simplify to slope-intercept form

Distribute the slope on the right side: \( y - 6 = -\frac{4}{3}x + 4 \). Add 6 to both sides to solve for \( y \): \( y = -\frac{4}{3}x + 4 + 6 \).

Step 5: Finalize the equation

Combine the constants on the right side: \( y = -\frac{4}{3}x + 10 \). This is the slope-intercept form of the line.

Key Concepts

Slope Calculation

To understand the concept of slope calculation, visualize it as measuring the steepness or incline of a line. When you hear "slope," think about how a line tilts across the coordinate plane. The slope is often represented by the letter "m."

The formula to calculate slope is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula helps us find out how much the "y" value changes for a given change in "x" value. You can think of the slope as "rise over run." It tells us how many units the line rises or falls vertically for every unit it increases horizontally.

In our exercise, with two points \(3,6\) and \(0,10\), the slope is calculated as follows:

• Identify each pair: \(x_1, y_1\) as \(3, 6\) and \(x_2, y_2\) as \(0, 10\).
• Plug values into the formula: \( m = \frac{10 - 6}{0 - 3} = -\frac{4}{3} \).
• The slope \( m \) is -\frac{4}{3}, indicating a downward tilt.

Using these steps, you can quickly find the slope for any two points on a line.

Point-Slope Form

The point-slope form is a handy tool for writing the equation of a line when you already know the slope and at least one point on the line. The format of this equation is:\[\ y - y_1 = m(x - x_1)\]This formula uses a specific point \(x_1, y_1\) and the slope "m" that you've calculated.

Let's see how this form works in our exercise. Given the slope \( m = -\frac{4}{3} \), and the point \(3, 6\):

• Plug \(3, 6\) and \( m = -\frac{4}{3} \) into the formula: \( y - 6 = -\frac{4}{3}(x - 3) \).
• This forms the equation of the line in point-slope format.

The point-slope form is particularly useful when you have distinct data points but need to derive a broader equation applicable across multiple values of "x."

It provides a straightforward path to shift to other forms like the slope-intercept equation.

Linear Equations

Linear equations are equations that form straight lines when plotted on a coordinate plane. A key characteristic of linear equations is their constant slope, meaning the rate of change between any two points on the line is always the same.

The slope-intercept form, which is a type of linear equation, is particularly easy to use and understand. Its form is:\[y = mx + b\]where:

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

In our exercise, we converted the point-slope form \(y - 6 = -\frac{4}{3}(x - 3)\) into slope-intercept form. Here's how:

• Distribute \(-\frac{4}{3}\) across \(x - 3\) to get \(y - 6 = -\frac{4}{3}x + 4\).
• Add 6 to both sides, resulting in \(y = -\frac{4}{3}x + 10\).
• Now, \( b = 10 \), which tells you that when \(x=0\), \(y\) will be 10.

Linear equations are a core mathematical concept because they can simplify complex data into a format that is easy to interpret and predict.