Problem 37 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Perpendicular to the line \(y=-\frac{2}{3}(x-1980)+5\) passing through \((1980,10)\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = \frac{3}{2}x - 2960\).

Step 1: Find the Slope of the Given Line

The given equation is already in slope-intercept form, but it's written slightly differently as \( y = -\frac{2}{3}(x-1980) + 5 \). The slope is \(-\frac{2}{3}\).

Step 2: Find the Perpendicular Slope

Lines that are perpendicular to each other have slopes that are negative reciprocals. The negative reciprocal of \(-\frac{2}{3}\) is \(\frac{3}{2}\). This will be the slope of our new line.

Step 3: Use Point-Slope Form

Using the point-slope form of the equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through and \(m\) is the slope. We know these values are: \(m = \frac{3}{2}\), \(x_1 = 1980\), \(y_1 = 10\). Substitute these into the formula: \(y - 10 = \frac{3}{2}(x - 1980)\).

Step 4: Simplify to Slope-Intercept Form

Distribute the \(\frac{3}{2}\) in the equation: \(y - 10 = \frac{3}{2}x - \frac{3}{2} \times 1980\). Calculating \(\frac{3}{2} \times 1980 = 2970\). This results in \(y - 10 = \frac{3}{2}x - 2970\). Add 10 to both sides to complete the equation: \(y = \frac{3}{2}x - 2960\).

Key Concepts

Perpendicular Lines

When two lines are perpendicular, they meet to form a right angle, which is 90 degrees. In the context of algebra and geometry, understanding the slopes of these lines is key.
For any two lines to be perpendicular, the product of their slopes should equal -1. This means that one slope is the negative reciprocal of the other.
For instance, if a line has a slope of \(-\frac{2}{3}\), the perpendicular line will have a slope of \(\frac{3}{2}\). This concept helps in finding explicit equations for lines that need to be perpendicular to existing lines.

Point-Slope Form

The point-slope form of a line's equation is essential when you know a line's slope and a specific point it passes through. It is expressed as: \(y - y_1 = m(x - x_1)\), where:

• \(m\) is the slope of the line,
• and \( (x_1, y_1)\) is the known point through which the line passes.

This formula serves as a bridge between information about a line and its graphical representation. Using it, you can describe lines precisely, leading directly to the slope-intercept form. When you substitute your known values into this form, you can derive an equation that defines the line in more common terms.

Negative Reciprocal

A negative reciprocal is a crucial concept for understanding perpendicular lines. To find the negative reciprocal of a number, you take the reciprocal (flip the number as represented in fraction form) and then change its sign.
For example, if you begin with \(-\frac{2}{3}\), to find its negative reciprocal means to first flip it to \(\frac{3}{2}\) and then change the sign to positive, resulting in \(\frac{3}{2}\).
This property is particularly useful in determining the slope of a line perpendicular to a given line. Knowing the slope as its negative reciprocal ensures that the lines form a 90-degree angle at the point where they meet.