Problem 119 - College Algebra Essentials

Question

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

Step 1: Find the Negative Reciprocal of the Given Slope

The negative reciprocal of \(-\frac{1}{4}\) is 4, as reciprocation changes the sign and makes \(-\frac{1}{4}\) become -4, but the negative makes it positive.

Step 2: Substitute the Negative Reciprocal and the Point into the Point Slope Form

Now substitute the point \((3,-5)\) and the negative reciprocal into y−y1=m(x−x1) to get the equation of the line. This results in y-(-5) = 4(x-3)

Step 3: Simplify the Equation

Simplify to y+5=4x-12.

Step 4: Transform the Equation into the General Form

Rearranging to get the equation in the general form (Ax +By = C), gives 4x-y=17

Key Concepts

Negative Reciprocal

When working with lines on a plane, the concept of the negative reciprocal plays a crucial role. Especially when it comes to perpendicular lines. The negative reciprocal is all about flipping a fraction and changing its sign. For instance, if you have a slope of \(-\frac{1}{4}\), the reciprocal would be \(-4\), but to find the negative reciprocal, you flip it and change the sign to get \4\. Therefore, if you multiply a slope by its negative reciprocal, the result is \(-1\).

• Original slope: \(-\frac{1}{4}\)
• Reciprocal: \(-4\)
• Negative reciprocal: \4\

This negative reciprocal relationship is also why perpendicular lines' slopes multiply to \(-1\). Recognizing and working with negative reciprocals is a foundational skill in geometry and algebra.

Point-Slope Form

The point-slope form is a handy way to write the equation of a line. If you know a point on the line and the slope, you can use this form. The format for point-slope form is:\[y - y_1 = m(x - x_1)\]Here, \(x_1, y_1\) is a point on the line, and \m\ is the slope. Suppose you're given the point \(3, -5\) and a slope of \4\, the point-slope form would be:\[y - (-5) = 4(x - 3)\]This format is highly useful for quickly generating a linear equation. After substituting your known values, it's a few quick steps to simplify it into other forms of a line equation.

• Allows for an immediate start by using a known point and slope.
• Simple substitution leads to ease in transformation to other forms.
• Helpful for visualizing lines with known values.

General Form

The general form of a line is a straightforward representation of a line's equation. It is structured as \(Ax + By = C\). This form does not specifically emphasize the slope or y-intercept, but it sets the equation in a standardized form, useful for many standard calculations and proofs.For instance, from the point-slope equation \(y + 5 = 4x - 12\), we rearrange terms to match the general form:\[4x - y = 17\]

• Standardized format across different mathematical contexts.
• Helps in solving systems of equations.
• Common format for programming and algorithmic purposes.

Using general form, it becomes easy to see and manipulate multiple equations at the same time, especially useful when finding intersections or understanding the relationship between multiple lines.