91影视

Hoshi drew graphs for \(y=x\) and \(y=-x\) and noticed that the lines crossed at right angles at the point \((0,0) .\) Then he drew graphs for \(y=x+4\) and \(y=-x+4\) and noticed that the lines crossed at right angles again, this time at the point \((0,4) .\) He tried one more pair, \(y=x-4\) and \(y=-x-4 .\) Once again the lines crossed at right angles, at the point \((-4,0)\). (Table not Copy) Hoshi made this conjecture: 鈥淲hen you graph two linear equations and one has a slope that is the negative of the other, you always get a right angle.鈥� a. Do you agree with Hoshi鈥檚 conjecture? Why or why not? b. Draw several more pairs of lines that fit the conditions of Hoshi鈥檚 conjecture, with different slope values. Do your drawings prove or disprove Hoshi鈥檚 conjecture? c. If you think Hoshi鈥檚 conjecture is false, where do you think he made his mistake?

Step by step solution

01

Understand the Conjecture

Hoshi's conjecture states that when two linear equations have slopes that are negative reciprocals of each other, they will always intersect at right angles.

02

Analyze Initial Examples

Review the given examples: 1. Lines: \( y = x \) and \( y = -x \) intersect at (0,0)2. Lines: \( y = x + 4 \) and \( y = -x + 4 \) intersect at (0,4)3. Lines: \( y = x - 4 \) and \( y = -x - 4 \) intersect at (-4,0)All these pairs of lines having slopes 1 and -1 intersect at right angles.

03

Justification of Right Angles

When one slope is the negative reciprocal of the other, the product of the slopes is -1. Algebraically, if line 1 has slope m and line 2 has slope -1/m, then:\[m \times \left(-\frac{1}{m}\right) = -1\]This means the lines are perpendicular and form a right angle at the point of intersection.

04

Test with Different Slope Values

Draw additional pairs of lines with slopes that are negative reciprocals of each other:1. Lines: \( y = 2x \) and \( y = -\frac{1}{2}x \)2. Lines: \( y = 3x + 1 \) and \( y = -\frac{1}{3}x + 1 \)3. Lines: \( y = -4x \) and \( y = \frac{1}{4}x \)Check if they intersect at right angles.

05

Verify with Drawings

By plotting these equations on a graph, it can be verified that the lines indeed intersect at right angles. This supports Hoshi's conjecture.

06

Conclusion on Conjecture

Based on the analysis and additional lines tested, Hoshi鈥檚 conjecture appears to be true as lines with slopes that have negative reciprocals always intersect at right angles. His conjecture is correct because the algebraic justification holds universally.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

linear equations

A linear equation is an equation that forms a straight line when graphed. The general form of a linear equation is: \( ax + by = c \) where:

1. \(a, b, \) and \(c\) are constants.
2. \(x\) and \(y\) are variables.

Linear equations are fundamental in coordinate geometry and algebra. They are simple relationships between two variables, usually \(x\) and \(y\), that can be represented graphically by a straight line. For example, \( y = 2x + 3 \) is a linear equation in slope-intercept form, which we will discuss in the next section.

slope-intercept form

Slope-intercept form is a way of writing a linear equation using the slope and the y-intercept. The general form is: \( y = mx + b \) where:

• \(m\) is the slope.
• \(b\) is the y-intercept.

The slope \(m\) represents the steepness or incline of the line, and the y-intercept \(b\) is the point where the line crosses the y-axis. For instance, for the equation \(y = mx + b\):

• If \(m = 2\), the line rises 2 units for every 1 unit run.
• If \(b = 3\), the line intersects the y-axis at \((0, 3)\).

Understanding slope-intercept form is crucial for solving and graphing linear equations.

right angles

A right angle is an angle of exactly 90 degrees. When two lines intersect at a right angle, they are said to be perpendicular. In the context of linear equations, if two lines intersect at a right angle, their slopes have a special relationship. This relationship involves negative reciprocals, which we will explore next. Perpendicular lines and right angles are fundamental concepts in geometry and are often used to verify the relationship between linear equations, such as in the exercise example where lines with slopes of 1 and -1 intersect at right angles.

negative reciprocals

Negative reciprocals are a key concept in understanding when two lines are perpendicular. Two numbers are negative reciprocals if their product is -1. For linear equations, this means: If line 1 has a slope of \(m\), then line 2 must have a slope of \(\frac{-1}{m}\) for them to be perpendicular. Algebraically, \[ m \times \left( -\frac{1}{m} \right) = -1 \]. This ensures that the two lines intersect at a right angle. For example, if one line has a slope of 2 (\(m = 2\)), then the perpendicular line must have a slope of \(\frac{-1}{2}\). Testing various pairs of lines with negative reciprocal slopes, as in Hoshi's exercise, confirms that they indeed form right angles, validating the conjecture. With this fundamental understanding, you can determine whether two lines will be perpendicular simply by examining their slopes.