/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Find the slope of each line and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 34

Find the slope of each line and a point on the line. Then graph the line. $$x=4-3 t, y=-2+6 t$$

Short Answer

Expert verified
Answer: The equation of the line in Cartesian form is $$y=-2x+6$$. The slope of the line is $$m=-2$$. A point on the line is (4, -2).

Step by step solution

01

Rewrite the given parametric equations into one Cartesian equation

Let's begin by solving the parametric equations for $$t$$. In the first equation, we have $$t = \frac{4-x}{3}$$. Now, substitute this value of $$t$$ into the second equation, we get: $$y=-2+6\left(\frac{4-x}{3}\right)$$
02

Find the slope

To find the slope of the line, we can rearrange the equation and find the derivative with respect to x. First, simplify the above equation: $$y=-2+8-2x$$ Now, we can rearrange the equation, putting it into the form $$y=mx+b$$, so that we can easily identify the slope. $$y=-2x+6$$ The slope, represented by the coefficient of the x-term, is $$m=-2$$.
03

Find a point on the line

Choose any value for $$t$$ and plug it into the parametric equations to find the coordinates of a point on the line. Let's assume $$t=0$$: $$x=4-3(0) = 4$$ $$y=-2+6(0) = -2$$ The point \((4, -2)\) lies on the line.
04

Graph the line

We have the equation in the form $$y=-2x+6$$, the slope $$m=-2$$, and a point on the line \((4, -2)\). When graphing the line, start at the point \((4,-2)\) and use the slope to find other points on the line. Since the slope is negative, the line will decrease as we move to the right. At each point, the line will have a rise of -2 and a run of 1.

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!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes, and directrices. Use a graphing utility to check your work. An ellipse with vertices (0,卤9) and eccentricity \(\frac{1}{4}\)

Sketch the graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$4 x=-y^{2}$$

Show that an ellipse and a hyperbola that have the same two foci intersect at right angles.

Explain and carry out a method for graphing the curve \(x=1+\cos ^{2} y-\sin ^{2} y\) using parametric equations and a graphing utility.

Find an equation of the line tangent to the following curves at the given point. $$y^{2}-\frac{x^{2}}{64}=1 ;\left(6,-\frac{5}{4}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.