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Chapter 1: Problem 32

Alejandro looked at the equations \(y=\frac{3}{2} x-1\) and \(y=-\frac{2}{3} x+2\) and said, These lines form a right angle. a. Graph both lines on two different grids, with the axes labeled as shown here. b. Compare the lines on each grid. Do both pairs of lines form a right angle? c. What kind of assumption must Alejandro have made when he said the lines form a right angle?

Short Answer

Expert verified
The lines form a right angle because their slopes multiply to \(-1\). Alejandro assumed correctly based on the slopes.

Step by step solution

01

Analyze the slopes of given equations

Identify the slopes of the given lines from the equations. The first equation is given by \(y = \frac{3}{2} x - 1\). Here, the slope \(m_1 = \frac{3}{2}\). The second equation is given by \(y = -\frac{2}{3} x + 2\). Here, the slope \(m_2 = -\frac{2}{3}\).
02

Determine the relationship between the slopes

For two lines to be perpendicular, the product of their slopes must be \(-1\). Let's calculate the product of the given slopes: \(\frac{3}{2} \times -\frac{2}{3} = -1\). Since the product is \(-1\), the two lines are perpendicular and form a right angle.
03

Graph the first line

On a graph, plot the line represented by the equation \(y = \frac{3}{2} x - 1\). This line has a slope of \(\frac{3}{2}\) and a y-intercept at \(-1\).
04

Graph the second line

On a separate graph, plot the line represented by the equation \(y = -\frac{2}{3} x + 2\). This line has a slope of \(-\frac{2}{3}\) and a y-intercept at \(2\).
05

Compare the lines on the graphs

Observe the graphs of the two lines. Since both pairs of lines have slopes that multiply to \(-1\), both pairs form a right angle.
06

Identify Alejandro's assumption

Alejandro must have assumed that the slopes of the lines determine whether or not they form a right angle. This is correct, as the product of the slopes of the given lines is \(-1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Slope
The slope of a line is a measure of its steepness. It is often represented by the letter 'm' in the linear equation form: y = mx + b. The slope indicates how much the 'y' value changes for a given change in the 'x' value. If a line has a positive slope, it goes upwards from left to right. If the slope is negative, the line goes downwards. For example:
  • If a line has a slope of \(\frac{3}{2}\), this means for every 2 units you move to the right, the line moves up by 3 units.
  • If a line has a slope of \( -\frac{2}{3} \), this means for every 3 units you move to the right, the line moves down by 2 units.
Understanding the slope is crucial for graphing linear equations and determining the relationship between different lines.
Graphing Linear Equations
Graphing linear equations involves plotting points on a grid to represent the equation 'y = mx + b'. Here are the steps you can follow:
  • Identify the slope 'm' and the y-intercept 'b' from the equation.
  • Start by plotting the y-intercept on the y-axis. For \(y = \frac{3}{2} x - 1\), plot the point (0, -1).
  • Use the slope to find another point. For a slope of \(\frac{3}{2}\), move 2 units to the right and 3 units up, plotting the point (2, 2).
  • Draw a line through the points.
Repeat these steps to graph different linear equations on separate grids. This makes it easier to compare the behavior and relationship of the lines.
The Concept of Right Angles
In geometry, a right angle is an angle of exactly 90 degrees. When two lines intersect to form a right angle, we say the lines are perpendicular. To determine if two lines form a right angle, you can check their slopes. The slopes of perpendicular lines have a special relationship: their product is always -1. For example, if the slope of one line is \(\frac{3}{2}\), the slope of a line perpendicular to it would be \(-\frac{2}{3}\) because:\[ \frac{3}{2} \times -\frac{2}{3} = -1 \]Seeing this product tells us conclusively that the lines are perpendicular and meet at a right angle.
Understanding the Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. It is represented by the 'b' in the equation 'y = mx + b'. The y-intercept is important because it gives you a starting point for graphing the line. For instance:
  • For the equation \(y = \frac{3}{2} x - 1\), the y-intercept is -1. This tells us the line crosses the y-axis at the point (0, -1).
  • For \(y = -\frac{2}{3} x + 2\), the y-intercept is 2. So the line crosses the y-axis at (0, 2).
To graph a line correctly, plot the y-intercept first, then use the slope to find another point on the line. From there, you can draw the line accurately.

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