Problem 47 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Passing through \((-2,4)\) and perpendicular to the line passing through \(\left(-5, \frac{1}{2}\right)\) and \(\left(-3, \frac{2}{3}\right)\)

Step-by-Step Solution

Answer

The slope-intercept form is \(y = -12x - 20\).

Step 1: Find the Slope of the Given Line

To find the slope of the line passing through the points \((-5, \frac{1}{2})\) and \((-3, \frac{2}{3})\), use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points, we get \(m = \frac{\frac{2}{3} - \frac{1}{2}}{-3 + 5}\). First, find a common denominator for the fractions: \(m = \frac{\frac{4}{6} - \frac{3}{6}}{2} = \frac{\frac{1}{6}}{2} = \frac{1}{12}\). Thus, the slope of the given line is \(m = \frac{1}{12}\).

Step 2: Find the Slope of the Perpendicular Line

A line that is perpendicular to another will have a slope that is the negative reciprocal of the other line's slope. Thus, the slope of the line perpendicular to \(\frac{1}{12}\) is \(-12\).

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

Using the point-slope form of a line equation \(y - y_1 = m(x - x_1)\), where \(m = -12\) and the point \((-2, 4)\), we substitute these values. This gives us \(y - 4 = -12(x + 2)\).

Step 4: Convert to Slope-Intercept Form

To convert the equation \(y - 4 = -12(x + 2)\) into slope-intercept form \(y = mx + b\), first distribute the \(-12\): \(y - 4 = -12x - 24\). Then, solve for \(y\) by adding \(4\) to both sides: \(y = -12x - 20\). This is the slope-intercept form.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a tool that makes it easy to write the equation of a line when you know a point on the line and its slope. It's expressed as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \((x_1, y_1)\) is a known point on the line. This form is named for the way it uses a specific point and the slope to create the equation.

Here's how it works:

• Start by identifying the slope \( m \). This tells you how steep the line is.
• Next, find a specific point on the line. The coordinates of this point are plugged in as \(x_1\) and \(y_1\).
• Substitute these values into the point-slope formula to find the equation of your line.

Using this form is particularly helpful when you're given a slope and a single point because it directly accommodates these two pieces of information without needing additional calculations. Once you have set up the equation in point-slope form, it's fairly straightforward to manipulate it into slope-intercept form \( y = mx + b \) if necessary.

Perpendicular Lines

Perpendicular lines have a special relationship when it comes to their slopes. If two lines are perpendicular, the product of their slopes is -1. This means the slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope.

To find the slope of a line that is perpendicular to a given line:

• First, determine the slope \( m \) of the original line.
• The perpendicular slope will be \(-\frac{1}{m}\).

For example, if a line has a slope of \( \frac{1}{12} \), the slope of a perpendicular line is \( -12 \). This is because you essentially "flip" the fraction and change the sign.

Understanding the relationship between perpendicular slopes is crucial in geometry, especially when you need to confirm right angles or construct perpendicular bisectors in different mathematical applications. This principle is widely used in various fields such as construction and design besides solving standard geometry problems.

Slope Calculation

The concept of slope reflects the steepness and direction of a line, which is key in understanding and graphing linear equations. The formula for calculating slope from two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula gives you the "rise over run," which measures how much the line goes up or down (rise) for a given horizontal change (run).

Here's how to use it:

• Subtract the \( y \)-coordinates to find the rise.
• Subtract the \( x \)-coordinates to find the run.
• Divide the rise by the run to find the slope \( m \).

This calculation can work with any two points on a line, making it a versatile approach for different problems. The slope helps determine the angle and orientation of the line, whether it rises or falls as it moves from left to right. This fundamental concept is critical in many areas of mathematics, physics, and even economics for analyzing trends and relationships between variables.