Problem 9 - College Algebra with Modeling and Visualization

Question

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. \(\mathbf{x}\) -intercept \(4, y\) -intercept \(-3\)

Step-by-Step Solution

Answer

The point-slope form is \(y - 0 = \frac{3}{4}(x - 4)\) and the slope-intercept form is \(y = \frac{3}{4}x - 3\).

Step 1: Identify two points

From the given intercepts, we know two points on the line: the x-intercept where the line crosses the x-axis, which is \((4, 0)\), and the y-intercept where the line crosses the y-axis, which is \((0, -3)\).

Step 2: Calculate the slope (m)

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the two points \((4, 0)\) and \((0, -3)\), calculate the slope:\[m = \frac{-3 - 0}{0 - 4} = \frac{-3}{-4} = \frac{3}{4}\,.\]

Step 3: Write the point-slope form

The point-slope form of a line is given by \(y - y_1 = m(x - x_1)\). Using the point \((4, 0)\) and the slope \(m = \frac{3}{4}\), the equation becomes:\[y - 0 = \frac{3}{4}(x - 4)\,.\]

Step 4: Simplify the point-slope form equation

Simplify the equation to\[y = \frac{3}{4}(x - 4)\,.\]

Step 5: Convert to slope-intercept form

To convert the equation to slope-intercept form \(y = mx + b\), distribute \(\frac{3}{4}\) across \((x - 4)\):\[y = \frac{3}{4}x - \frac{3}{4}(4)\ = \frac{3}{4}x - 3\,.\]

Step 6: Confirm the equation

The slope-intercept form of the line is \(y = \frac{3}{4}x - 3\). This matches the given x- and y-intercepts when x is 4 (y=0) and when x is 0 (y=-3).

Key Concepts

Slope

The slope is a crucial characteristic of a line in geometry and algebra, representing how steep the line is. It tells us how much the y-value of a line increases or decreases as the x-value changes.
In mathematical terms, the slope (\( m \)) is calculated using two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), with the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula indicates the change in y divided by the change in x.

• If the slope is positive, the line ascends from left to right.
• If it's negative, the line descends.
• A zero slope means the line is horizontal, while an undefined slope means it's vertical.

The slope of a line can often tell you much about the line's identity and angle.

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular format in algebra due to its simplicity. It is represented as \( y = mx + b \).
This form allows us to easily identify two vital properties of the line: the slope \( m \) and the y-intercept \( b \).
Here's what each component represents:

• \( m \): the slope of the line, giving insight into its steepness.
• \( b \): the y-intercept, showing the point where the line crosses the y-axis.

Let's consider the line from our original problem statement which converts to this form as: \( y = \frac{3}{4}x - 3 \).
From this equation:

• \( m = \frac{3}{4} \) indicates that for every 4 units increase in x, y increases by 3 units.
• \( b = -3 \) shows that the line crosses the y-axis at -3.

This form is especially handy for graphing linear equations quickly.

Linear Equation

A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. These equations are fundamental in algebra and are straightforward to work with due to their constant rate of change.
The general formula for a linear equation can be written as \( Ax + By = C \), and it is closely associated with the familiar slope-intercept form \( y = mx + b \).

Here are some important characteristics:

• Coefficients \( A, B, \) and constant \( C\) are real numbers.
• The highest power of the variable is 1.

These properties make linear equations predictable and easy to interpret.
For example, our exercise dealt with a line crossing the x-axis at 4 and the y-axis at -3. This type of equation reliably outlines such a line's behavior. Understanding linear equations is not only crucial in academic quizzes but also in various real-world applications, from calculating distances to predicting business sales trends.