Problem 48 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Passing through \(\left(\frac{3}{4}, \frac{1}{4}\right)\) and perpendicular to the line passing through \((-3,-5)\) and \((-4,0)\)

Step-by-Step Solution

Answer

The slope-intercept form is \( y = \frac{1}{5}x + \frac{1}{10} \).

Step 1: Find Slope of Given Line

To find the slope of the line passing through \((-3,-5)\) and \((-4,0)\), use the slope formula:\[ m = \frac{y_2-y_1}{x_2-x_1} \]Substituting the given points, we have \[ m = \frac{0 - (-5)}{-4 - (-3)} = \frac{5}{-1} = -5 \] So, the slope of the given line is \(-5\).

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. For a slope of \(-5\), the perpendicular slope is \[ m = \frac{1}{5} \] Thus, the slope of the line we are looking for is \(\frac{1}{5}\).

Step 3: Use the Point-Slope Form to Find the Equation

We use the point-slope form of a line: \[ y - y_1 = m(x - x_1) \]Substitute \(m = \frac{1}{5}\), \(x_1 = \frac{3}{4}\), and \(y_1 = \frac{1}{4}\) from the point \(\left(\frac{3}{4}, \frac{1}{4}\right)\):\[ y - \frac{1}{4} = \frac{1}{5}(x - \frac{3}{4}) \]

Step 4: Simplify to Slope-Intercept Form

Expand and simplify the equation \[ y - \frac{1}{4} = \frac{1}{5}x - \frac{3}{20} \]Add \(\frac{1}{4}\) to both sides to isolate \(y\):\[ y = \frac{1}{5}x - \frac{3}{20} + \frac{5}{20} \] Combine the fractions on the right:\[ y = \frac{1}{5}x + \frac{2}{20} \] Simplify further:\[ y = \frac{1}{5}x + \frac{1}{10} \]The slope-intercept form is \[ y = \frac{1}{5}x + \frac{1}{10} \].

Key Concepts

Perpendicular Lines

When two lines intersect at right angles, they are said to be perpendicular. This means they form a 90-degree angle with each other. In the context of the slope of these lines, there's a special relationship between perpendicular lines. If the slope of one line is known, you can easily find the slope of a line perpendicular to it.
The crucial rule to remember is that the slopes of two perpendicular lines are negative reciprocals of each other.
Here's how you can use this:

• If the slope of one line is denoted by \( m \), the slope of the line perpendicular to it will be \(-\frac{1}{m}\).
• This relationship is essential in geometry, often used in constructing perpendicular bisectors or finding equations of lines in algebra.

In the given exercise, the slope of the original line was \(-5\). Thus, the slope of the perpendicular line we used was \(\frac{1}{5}\), following this negative reciprocal rule.

Point-Slope Form

The point-slope form of a linear equation is a powerful tool used to find the equation of a line. Unlike the more general slope-intercept form \( y = mx + b \), the point-slope form focuses directly on a known point and the slope.
This form is written as:

• \( y - y_1 = m(x - x_1) \)

where \( m \) is the slope, and \((x_1, y_1)\) is a specific point the line passes through.
Point-slope form is particularly convenient when you have a point and the slope ready, as it allows for quick calculations:

• You can use it to immediately plug in these values, generating an equation for the line.
• It's a great tool for transitioning to the slope-intercept form if needed, simplifying your process further.

In the presented problem, we utilized \(\left(\frac{3}{4}, \frac{1}{4}\right)\) for our point and \(\frac{1}{5}\) as our slope, directly applying these into the point-slope formula to find the needed equation.

Slope Calculation

Calculating the slope of a line is a fundamental skill in algebra and geometry. It tells you how steep a line is, or in practical terms, how much \( y \) changes for a given change in \( x \). The slope formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This uses two points, \((x_1, y_1)\) and \((x_2, y_2)\), showing the change in \( y \) over the change in \( x \).
Let's go step by step:

• Subtract the second \( y \) value from the first to find the change in the vertical direction (\( y_2 - y_1 \)).
• Subtract the second \( x \) value from the first to find the change in the horizontal direction (\( x_2 - x_1 \)).

Then, divide these results to get your slope. In the exercise, using points \((-3, -5)\) and \((-4, 0)\), the calculation was straightforward, resulting in a slope of \(-5\). This value is key to understanding the relationship and behavior of the line, especially when seeking its perpendicular counterpart.