Problem 33 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line passing through the given point and perpendicular to the given line. $$ (0,3), y=\frac{7}{8} x $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the point (0,3) and is perpendicular to the given line \(y=\frac{7}{8}x\) is \(y = -\frac{8}{7}x + 3\).

Step 1: Find the Slope \(m'\)

The slope of the line perpendicular to the given line is the negative reciprocal of the given line's slope. This is calculated as \(m' = -1/m = -1/(\frac{7}{8}) = -\frac{8}{7}.\)

Step 2: Identify the y-intercept \(b'\)

The y-intercept is given by the y-coordinate of the point that the line passes through. Here, the line passes through the point (0,3), so our y-intercept \(b'\) is 3.

Step 3: Substitute into the Slope-Intercept Form

Finally, substitute the slope \(m'\) and the y-intercept \(b'\) into the equation \(y = mx + b\) to get the equation of the line in slope-intercept form. This gives the final equation: \(y = -\frac{8}{7}x + 3\).

Key Concepts

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). Understanding perpendicularity in mathematics is crucial when working with line equations. Two lines are perpendicular if the product of their slopes is -1. This special relationship allows you to determine whether two lines will meet at a right angle without having to graph them.

Imagine two lines on a graph: when they cross each other forming an "L" or "T" shape, they are perpendicular. Identifying perpendicular lines is a key skill in geometry and algebra.

Negative Reciprocal

The concept of negative reciprocal is essential when dealing with perpendicular lines. If you know the slope of a line, you can easily find the slope of a line that is perpendicular to it by taking the negative reciprocal.

Simply put, the negative reciprocal of a number is the opposite sign and flipped fraction of that number. For example, the negative reciprocal of \( \frac{7}{8} \) is \( -\frac{8}{7} \). This relationship ensures that when the slopes are multiplied together, they result in -1:

• If the slope of one line is \( m \), the slope of a line perpendicular to it will be \( -\frac{1}{m} \).
• This concept helps maintain the right-angle intersection between perpendicular lines.

Equation of a Line

The equation of a line in slope-intercept form is vital in expressing linear relationships. It is generally written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept, or the point where the line crosses the y-axis.

This format allows you to quickly understand the steepness and position of a line on a graph.

To write the equation of a line:

• Determine the slope \( m \), which indicates how much \( y \) will increase or decrease as \( x \) increases by 1.
• Identify the y-intercept \( b \), which shows where the line crosses the y-axis.
• Combine these into the equation \( y = mx + b \) to describe the line completely.

This simple yet powerful tool allows you to represent and predict relationships graphically and numerically.