Problem 122 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. Write an equation in general form of the line passing through (3,-5) whose slope is the negative reciprocal (the reciprocal with the opposite sign) of \(-\frac{1}{4}\)

Step-by-Step Solution

Answer

The equation of the line in general form is \(4x - y - 17 = 0\).

Step 1: Calculate the Slope

The negative reciprocal of \(-\frac{1}{4}\) is calculated by flipping the fraction and changing the sign. Thus, the negative reciprocal is \(4\). This means the slope \(m\) of our line is \(4\).

Step 2: Use the Slope-Point Form

The slope-point form of a line is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope of the line. Insert the point (3, -5) and the slope \(4\) into this formula: \(y - (-5) = 4(x - 3)\). This simplifies to \(y + 5 = 4x - 12\).

Step 3: Convert to General Form

The general form of a line is \(Ax + By + C = 0\). Convert the equation from Step 2 to this form by subtracting \(y\) and \(5\) from both sides, which gives the equation \(4x - y - 17 = 0\).

Key Concepts

General Form

The general form of a linear equation is an important structure in algebra. It is a way to express a line using three coefficients and is written as \( Ax + By + C = 0 \). In this form, \( A \), \( B \), and \( C \) are constants, and both \( x \) and \( y \) are variables.

• It provides a standardized way to represent lines, making it easier to manipulate equations or analyze them.
• This form is particularly useful for determining intersections, parallels, and perpendiculars between lines.

To convert any linear equation to general form, you must rearrange it so that all terms are on one side of the equation, usually with \( x \) first and \( y \) second. For example, starting from the slope-intercept form \( y = mx + b \), you can subtract \( y \) from both sides, resulting in \( mx - y + b = 0 \). Here, \( m \) becomes \( A \), \( -1 \) becomes \( B \), and \( b \) is \( C \). While working with equations in general form, ensure that \( A \), \( B \), and \( C \) are integers, as it is standard practice.

Slope-Point Form

The slope-point form of a line is a very handy way to find the equation of a line when you know its slope and a point on the line. The formula is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) is a specific point through which the line passes.

• This form is incredibly useful, especially in cases where you need to convert from a simple known point and slope into an equation quickly.
• It allows you to immediately visualize the slope through the coefficient \( m \) and define the line's location through the point \((x_1, y_1)\).

To use the slope-point form, you substitute the known values of the point and slope into the equation. For instance, if a line has a slope of \( 4 \) and passes through the point \( (3, -5) \), substituting these values gives you the equation \( y - (-5) = 4(x - 3) \), which simplifies to \( y + 5 = 4x - 12 \). From there, it's easy to rearrange into other forms, like the general form.

Negative Reciprocal

Understanding the concept of a negative reciprocal is key in geometry when dealing with perpendicular lines and slopes. The negative reciprocal is calculated by taking the reciprocal of a number and then changing its sign.

• For instance, the negative reciprocal of \(-\frac{1}{4}\) is \(4\). Flip the fraction to \(\frac{4}{1}\) and change the sign from negative to positive.
• The concept is widely used when determining the slope of a line perpendicular to another since the product of the slopes of two perpendicular lines in a plane is \(-1\).

To find a negative reciprocal, simply invert the fraction and multiply by \(-1\). For instance, if you begin with \(-\frac{3}{2}\), the negative reciprocal would be \(\frac{2}{3}\). It's a simple but powerful tool, particularly in solving problems where angles between lines are important, such as in the original exercise, it helped find the slope of the line by flipping and negating the reciprocal of an existing slope.