/*! 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 15 Find a number \(t\) such that th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 15

Find a number \(t\) such that the point \((3, t)\) is on the line containing the points (7,6) and (14,10) .

Short Answer

Expert verified
The number \(t\) such that the point \((3, t)\) is on the line containing the points (7,6) and (14,10) is \(t = \frac{26}{7}\).

Step by step solution

01

Find slope of the line passing through the points (7,6) and (14,10)

To find the slope of the line, use the formula for finding the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) which is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] In this case, \((x_1, y_1) = (7, 6)\) and \((x_2, y_2) = (14, 10)\). Therefore, the slope can be calculated as: \[m = \frac{10 - 6}{14 - 7} = \frac{4}{7}\]
02

Find the equation of the line

Now that we have the slope, we can use the slope-point form of the equation of a line which is: \[y - y_1 = m(x - x_1)\] Substitute the slope \(m = \frac{4}{7}\) and one of the given points, say \((x_1, y_1) = (7, 6)\), into the formula: \[y - 6 = \frac{4}{7}(x - 7)\]
03

Simplify the equation of the line

Now, simplify the equation further by distributing the slope on the right side and then isolating \(y\): \[\begin{aligned} y - 6 &= \frac{4}{7}(x - 7) \\ y &= \frac{4}{7}(x - 7) + 6 \end{aligned}\]
04

Substitute the x-coordinate of the point (3,t) and solve for t

Now, we'll substitute the x-coordinate from the point \((3, t)\) into the equation of the line and solve for \(t\): \[\begin{aligned} t &= \frac{4}{7}(3 - 7) + 6 \\ t &= \frac{4}{7}(-4) + 6 \\ t &= -\frac{16}{7} + 6 \end{aligned}\]
05

Convert the mixed number to a fraction and simplify

Finally, convert the mixed number to a fraction and simplify: \[\begin{aligned} t &= -\frac{16}{7} + \frac{42}{7} \\ t &= -\frac{16 - 42}{7} \\ t &= \frac{26}{7} \end{aligned}\] So the point \((3, t)\) is on the line containing the points (7,6) and (14,10) when \(t = \frac{26}{7}\).

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

Show that if \(p\) and \(q\) are nonzero polynomials, then $$ \operatorname{deg}(p \circ q)=(\operatorname{deg} p)(\operatorname{deg} q) $$.

A new snack shop on campus finds that the number of students following it on Twitter at the end of each of its first five weeks in business is 23,89,223 , \(419,\) and \(647 .\) A clever employee discovers that the number of students following the new snack shop on Twitter after \(w\) weeks is \(p(w),\) where \(p\) is defined by $$p(w)=7+3 w+5 w^{2}+9 w^{3}-w^{4}$$ Indeed, with \(p\) defined as above, we have \(p(1)=23,\) \(p(2)=89, p(3)=223, p(4)=419,\) and \(p(5)=647\) Explain why the polynomial \(p\) defined above cannot give accurate predictions for the number of followers on Twitter for weeks far into the future.

Show that $$ (a+b)^{3}=a^{3}+b^{3} $$ if and only if \(a=0\) or \(b=0\) or \(a=-b\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r-s)(x) $$

Write the indicated expression as \(a\) polynomial. $$ (p(x))^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.