Problem 56 - A Graphical Approach to College Algebra

Question

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 of the line is \( y = \frac{1}{5}x + \frac{1}{10} \).

Step 1: Find the Slope of the Given Line

The first task is to determine the slope of the line passing through \((-3, -5)\) and \((-4, 0)\). Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-5)}{-4 - (-3)} = \frac{5}{-1} = -5 \] Hence, the slope of the given line is \(-5\).

Step 2: Determine the Slope of the Perpendicular Line

For two lines to be perpendicular, the product of their slopes must be \(-1\). If the slope of the given line is \(-5\), let the slope of the required perpendicular line be \(m\). Thus, \(m \times (-5) = -1\). Solving for \(m\), we get \[ m = \frac{1}{5} \]So, the slope of the perpendicular line is \(\frac{1}{5}\).

Step 3: Write the Equation in Point-Slope Form

We have the slope of the perpendicular line as \(\frac{1}{5}\) and it passes through the point \(\left(\frac{3}{4}, \frac{1}{4}\right)\). Use the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] Substitute \(y_1 = \frac{1}{4}\), \(x_1 = \frac{3}{4}\), and \(m = \frac{1}{5}\) into the formula: \[ y - \frac{1}{4} = \frac{1}{5}(x - \frac{3}{4}) \]

Step 4: Simplify the Equation

Distribute the slope on the right side of the equation and simplify: \[ y - \frac{1}{4} = \frac{1}{5}x - \frac{1}{5} \times \frac{3}{4} \] \[ y - \frac{1}{4} = \frac{1}{5}x - \frac{3}{20} \]Add \(\frac{1}{4}\) to both sides to get the equation in slope-intercept form:\[ y = \frac{1}{5}x - \frac{3}{20} + \frac{5}{20} \] \[ y = \frac{1}{5}x + \frac{2}{20} \]Thus, the equation of the line is \[ y = \frac{1}{5}x + \frac{1}{10} \]

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in coordinate geometry, describing how steep a line is. It tells us how much the line rises or falls as we move from one point to another on the line.
The formula for finding the slope, \( m \), between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

The numerator \( (y_2 - y_1) \) represents the vertical change, while the denominator \( (x_2 - x_1) \) represents the horizontal change. If the slope is positive, the line rises as it moves from left to right; if negative, it falls.
In the original exercise, the slope of the line passing through points \((-3, -5)\) and \((-4, 0)\) was calculated as \(-5\). This indicates that for each unit we move to the right, the line descends 5 units down. Understanding slopes helps in recognizing various line properties and parallelism or perpendicularity of lines in geometry.

Point-Slope Form

The point-slope form of a line's equation is particularly useful when you know a point on the line and its slope. It provides a convenient way to write the equation of a line.
The formula for the point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is a point the line passes through, and \(m\) is the slope of the line. This form is ideal for quickly establishing the line's equation before potentially transforming it into other forms like the slope-intercept form.
In the given problem, after finding the perpendicular slope of \( \frac{1}{5} \), and given the point \( \left(\frac{3}{4}, \frac{1}{4}\right) \), we use the point-slope form to construct the line's equation. Plugging \( x_1 \), \( y_1 \), and \( m \) into the formula helps us build the direct relationship between x and y for this specific line.

Perpendicular Lines

Perpendicular lines intersect at right angles, a feature that is key in many geometric constructions and real-world applications, such as carpentry and city planning. The relationship between the slopes of two perpendicular lines in a plane is a handy trick in coordinate geometry.
If you know one line's slope, say \( m_1 \), the slope of the line perpendicular to it, \( m_2 \), fulfills the condition:

• \( m_1 \times m_2 = -1 \)

This relationship arises because the product of slopes for perpendicular lines is always \(-1\). It's an intuitive result when you visualize two lines meeting perfectly at a square corner.
In our problem, the given line has a slope of \(-5\). So to find a perpendicular line, you'd solve \(-5 \times m_2 = -1\). Fried from this equation, \( m_2 \) turns out to be \( \frac{1}{5} \). Knowing this enables the formation of the new line equation easily, which is crucial in contextual problems where angles and orientations are essential constraints.