Problem 36 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \(\left(\frac{1}{2},-\frac{3}{3}\right);\) perpendicular to the line \(4 x-8 y=1\)

Step-by-Step Solution

Answer

The equation is \( y = -2x \).

Step 1: Find the Slope of the Original Line

To find the slope of the given line, rewrite it in slope-intercept form, which is \( y = mx + b \). Start with the given line equation \( 4x - 8y = 1 \). Solve for \( y \).\[8y = 4x - 1 \y = \frac{4}{8}x - \frac{1}{8} = \frac{1}{2}x - \frac{1}{8}\]The slope \( m \) of the original line is \( \frac{1}{2} \).

Step 2: Find the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Here, the original slope is \( \frac{1}{2} \), so the perpendicular slope \( m \) is:\[m = -\frac{1}{\left(\frac{1}{2}\right)} = -2\]

Step 3: Use Point-Slope Form to Write the Equation

Now that we have the slope \( m = -2 \) for the perpendicular line, and a point \( \left(\frac{1}{2}, -1\right) \) that it passes through (note that \(-\frac{3}{3} = -1\)), we use the point-slope form \( y - y_1 = m(x - x_1) \):\[y - (-1) = -2 \left( x - \frac{1}{2} \right)\]Simplifying, this gives:\[y + 1 = -2x + 1\]

Step 4: Simplify to Slope-Intercept Form

Continue by simplifying and solving for \( y \) to find the equation in slope-intercept form:\[y = -2x + 1 - 1 \y = -2x\]

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line so you can easily identify the line's slope and where it intersects the y-axis. The general formula for the slope-intercept form is: \[ y = mx + b \] where:

• \( m \) is the slope of the line. It tells you how steep the line is.
• \( b \) is the y-intercept. It is the point where the line crosses the y-axis.

To convert a line equation into this form, you need to solve for \( y \). Once in this form, it's simple to determine the slope and y-intercept directly from the equation. For example, by rewriting the line equation from the problem, \( 4x - 8y = 1 \), into slope-intercept form, we find: \[ y = \frac{1}{2}x - \frac{1}{8} \] From this, we can see that the slope \( m \) is \( \frac{1}{2} \), and the line crosses the y-axis at \( -\frac{1}{8} \). This setup helps in analyzing and graphing lines with ease.

Point-Slope Form

The point-slope form is useful for writing the equation of a line when you know the slope and one point on the line. The formula is: \[ y - y_1 = m(x - x_1) \] where:

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

This form focuses on how the line behaves from a specific point. It's particularly handy when constructing a new line, like a line perpendicular to another line as in the problem. In our exercise, we used the point \( \left(\frac{1}{2}, -1\right) \) and the perpendicular slope \(-2\). Putting these into the point-slope formula gave us:\[ y - (-1) = -2 (x - \frac{1}{2}) \] This step bridges between initial information (point and slope) and the final line equation, emphasizing the flexibility of algebraic manipulation in geometry.

Negative Reciprocal

Understanding negative reciprocal is key when working with perpendicular lines. When two lines are perpendicular, their slopes are negative reciprocals of each other. To find the negative reciprocal of a slope, you:

• Invert the slope (flip the numerator and denominator).
• Change the sign (positive to negative or negative to positive).

For instance, in our exercise, the original line's slope was \( \frac{1}{2} \). To find the perpendicular slope, we take its negative reciprocal: \[ m = -\frac{1}{\left(\frac{1}{2}\right)} = -2 \] This concept ensures that the two lines meet at a right angle, forming the basis for solving many geometric problems involving perpendicularity. Being comfortable with negative reciprocals will enhance your ability to manage and solve problems involving perpendicular lines.