Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. 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 equation is \(y = \frac{1}{5}x + \frac{1}{10}\).

Step 1: Find the slope of the given line

To find the slope of the line passing through the points \((-3,-5)\) and \((-4,0)\), use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the coordinates: \(m = \frac{0 - (-5)}{-4 - (-3)} = \frac{5}{-1} = -5\). Thus, 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 original slope. Since the slope of the given line is \(-5\), the slope of the line perpendicular to it is \(\frac{1}{5}\).

Step 3: Use the point-slope form to find the equation

Use the point-slope form \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Substitute \(m = \frac{1}{5}\) and the point \(\left(\frac{3}{4}, \frac{1}{4}\right)\) into the formula: \(y - \frac{1}{4} = \frac{1}{5}(x - \frac{3}{4})\).

Step 4: Solve for slope-intercept form

Solve the equation \(y - \frac{1}{4} = \frac{1}{5}(x - \frac{3}{4})\) for \(y\). First distribute \(\frac{1}{5}\): \(y - \frac{1}{4} = \frac{1}{5}x - \frac{3}{20}\). Add \(\frac{1}{4}\) to both sides: \(y = \frac{1}{5}x - \frac{3}{20} + \frac{5}{20}\). Simplifying, \(y = \frac{1}{5}x + \frac{2}{20}\) or \(y = \frac{1}{5}x + \frac{1}{10}\). Thus, the slope-intercept form is \(y = \frac{1}{5}x + \frac{1}{10}\).

Key Concepts

Perpendicular Lines

Understanding perpendicular lines is crucial in geometry and algebra. Two lines are considered perpendicular if they intersect at a right angle (90 degrees).

This relationship can also be described in terms of their slopes:

• When two lines are perpendicular, the slope of one line is the negative reciprocal of the other.
• The term "negative reciprocal" means that if the slope of the first line is a fraction \( -\frac{a}{b} \), the slope of the line perpendicular to it would be \( \frac{b}{a} \).

For example, if a line has a slope of \(-5\), as given in our exercise, the slope of a line that is perpendicular to it will be \(\frac{1}{5}\). This relationship ensures the two lines meet at a right angle.

Slope Calculation

Calculating the slope of a line is a fundamental skill in algebra. The slope measures the steepness and direction of the line, often denoted by the letter \(m\).

To calculate the slope of a line passing through two distinct points, \((x_1, y_1)\) and \((x_2, y_2)\), you can use the following formula:

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

This formula calculates the change in \(y\) values divided by the change in \(x\) values.

For instance, using the points \((-3, -5)\) and \((-4, 0)\) from our example, substitute into the formula: \( m = \frac{0 - (-5)}{-4 - (-3)} = -5 \).

Hence, the slope of this line is \(-5\). This is a straightforward calculation essential for determining the nature of the line.

Point-Slope Form

The point-slope form is a useful formula for describing a line when you know its slope and a point on the line. It is represented as:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. This form is especially handy when you need to quickly write the equation of the line.

In the given example, the line passes through \(\left(\frac{3}{4}, \frac{1}{4}\right)\) and has a slope of \(\frac{1}{5}\) (perpendicular to the given line). Substituting these values into the formula gives: \(y - \frac{1}{4} = \frac{1}{5}(x - \frac{3}{4})\).

Solving for \(y\) will yield the equation in slope-intercept form. This is an essential process for finding equations of lines efficiently.