Problem 55 - A Graphical Approach to College Algebra

Question

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 equation 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 formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in the coordinates, we have: \[ m = \frac{\frac{2}{3} - \frac{1}{2}}{-3 + 5} = \frac{\frac{4}{6} - \frac{3}{6}}{2} = \frac{\frac{1}{6}}{2} = \frac{1}{12}. \] Thus, the slope of the given line is \( \frac{1}{12} \).

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Since the slope of the original line is \( \frac{1}{12} \), the slope of the perpendicular line is \(-12\).

Step 3: Write the Equation of the Perpendicular Line

Use the point-slope form of the equation, \( y - y_1 = m(x - x_1) \), where \((x_1, y_1) = (-2, 4)\), and \(m = -12\). Substitute these values into the equation: \[ y - 4 = -12(x + 2). \] Simplify to get the equation in slope-intercept form: \( y = -12x - 24 + 4 \), thus \( y = -12x - 20 \).

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is essential in coordinate geometry. The slope is a measure of the steepness and direction of a line. It is calculated by determining the change in the y-values divided by the change in the x-values between two points on the line. Mathematically, it's expressed as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here,

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
• \( m \) is the slope of the line.

For instance, if you have points like \((-5, \frac{1}{2})\) and \((-3, \frac{2}{3})\), calculate the slope using these coordinates. Calculate the difference in the y-values: \( \frac{2}{3} - \frac{1}{2} = \frac{1}{6} \). Next, calculate the difference in the x-values, \(-3 + 5 = 2 \). So the slope is: \( \frac{1}{12} \). Breaking down these calculations helps with understanding each step.

Point-Slope Form

After finding the slope, the point-slope form comes in handy for writing the equation of the line. It's perfect for quick transformations from points and slopes into line equations. The general formula for point-slope form is:\[ y - y_1 = m(x - x_1) \]Where:

• \((x_1, y_1)\) is a specific point on the line, and
• \(m\) is the slope of the line.

To write the equation of a line that is perpendicular to a given line, you first need the point it passes through and the slope. For example, given a point \((-2, 4)\) and a calculated perpendicular slope of \(-12\), substitute these values into the formula. This results in: \[ y - 4 = -12(x + 2) \]This representation gives you a "starting point" to further simplify into other forms, like the slope-intercept form.

Slope-Intercept Form

The slope-intercept form of a line equation is one of the most common forms used because it easily identifies the slope and the y-intercept of the line. The equation is presented as:\[ y = mx + b \]Where:

• \(m\) is the slope,
• \(b\) is the y-intercept.

In order to convert from point-slope to slope-intercept form, you need to solve for \(y\). Continuing with our example from the previous section where we had: \[ y - 4 = -12(x + 2) \]Distribute and simplify:\[ y = -12x - 24 + 4 \]So, the slope-intercept form of this line is:\[ y = -12x - 20 \]This clear format tells us that for every unit the line moves horizontally, it moves twelve units vertically. Meanwhile, it crosses the y-axis at \(-20\). This makes visualizing and graphing the line much easier!