/*! 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 89 Suppose \((2,5),(4,13),\) and \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 89

Suppose \((2,5),(4,13),\) and \((7, y)\) all lie on the same line. Find \(y .\)

Short Answer

Expert verified
The value of \(y\) is 25.

Step by step solution

01

Calculate the Slope

The first step is to calculate the slope of the line using the given points \((2, 5)\) and \((4, 13)\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the points into the formula:\[m = \frac{13 - 5}{4 - 2} = \frac{8}{2} = 4\]So, the slope \(m\) is 4.
02

Use the Point-Slope Form

Next, use the point-slope form of a line, which is:\[y - y_1 = m(x - x_1)\]We can use either of the given points, but let's use \((2, 5)\). Substitute \((x_1, y_1) = (2, 5)\) and \(m = 4\) into the equation:\[y - 5 = 4(x - 2)\]
03

Solve for y with x = 7

Finally, substitute \(x = 7\) (from the point \((7, y)\)) into the equation from Step 2 to find \(y\):\[y - 5 = 4(7 - 2)\]\[y - 5 = 4 \times 5\]\[y - 5 = 20\]Add 5 to both sides:\[y = 25\]
04

Verification Step: Check Consistency

To verify, we can substitute all points back into the equation obtained in Step 2:For \((2, 5)\):\[5 - 5 = 4(2 - 2)\Rightarrow 0 = 0\] For \((4, 13)\):\[13 - 5 = 4(4 - 2)\Rightarrow 8 = 8\]For \((7, 25)\):\[25 - 5 = 4(7 - 2)\Rightarrow 20 = 20\]All points satisfy the equation, confirming that our solution is correct.

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.

Slope Calculation
To understand how to find the slope of a line, think of it as a way to measure how steep the line is. The slope is given by the ratio of the change in the y-coordinates to the change in the x-coordinates between any two points on the line.

For our exercise, to calculate the slope using the points \((2,5)\) and \((4,13)\), we can use the formula:
  • Slope \(m\) = \( \frac{y_2 - y_1}{x_2 - x_1} \)
Substituting the values:
  • \(m = \frac{13 - 5}{4 - 2} = \frac{8}{2} = 4\)
This means the slope \(m\) is 4, indicating for every 1 unit increase in x, y increases by 4 units.
Point-Slope Form
The point-slope form is especially useful for writing the equation of a line when you know the slope and any point on the line. The standard formula is:
  • \( y - y_1 = m(x - x_1) \)
This equation enables us to plug in any point-specific values to model the line. Let's take the known point \((2, 5)\) and the slope \(4\):
  • \( y - 5 = 4(x - 2) \)
This is the core equation for the line containing the points. It can be re-arranged to find any y for a given x value by plugging that x into the equation.
Equation Verification
Verifying your equation is a crucial check to ensure its correctness. This involves substituting the points into the equation to see if both sides match. Let's verify the equation \( y - 5 = 4(x - 2) \) works for all points:
  • With \((2,5)\): \( 5 - 5 = 4(2 - 2) \rightarrow 0 = 0 \)
  • With \((4,13)\): \( 13 - 5 = 4(4 - 2) \rightarrow 8 = 8 \)
  • With \((7,25)\): \( 25 - 5 = 4(7 - 2) \rightarrow 20 = 20 \)
Each point checks out, confirming the equation's accuracy. When all given points satisfy the line equation, it indicates that the interpretations and calculations were done correctly, giving you confidence in your solution.

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

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. $$ f(x)=x^{4}+4 x^{3}-36 x^{2}-160 x+300 $$

Solve for \(y\) in terms of \(x\), and determine if the resulting equation represents a function. $$ \left(4 y^{2 / 3}\right)^{3}=64 x $$

Determine the domain of each function. $$ f(x)=\frac{x^{2}-4}{x+2} $$

In psychology a process called scaling is used to attach numerical ratings to a group of life experiences. In the following table, various events have been rated on a scale from 1 to 100 according to their stress levels. $$ \begin{array}{|l|c|} \hline \text { EVENT } & \text { SCALE OF IMPACT } \\ \hline \text { Death of spouse } & 100 \\ \text { Divorce } & 73 \\ \text { Jail term } & 63 \\ \text { Marriage } & 50 \\ \text { Lost job } & 47 \\ \text { Pregnancy } & 40 \\ \text { Death of close friend } & 37 \\ \text { Loan over } \$ 10,000 & 31 \\ \text { Child leaving home } & 29 \\ \text { Change in schools } & 20 \\ \text { Loan less than } \$ 10,000 & 17 \\ \text { Christmas } & 12 \\ \hline \end{array} $$ a) Does the table represent a function? Why or why not? b) What are the inputs? What are the outputs?

The Technology Connection heading indicates exercises designed to provide practice using a graphing calculator. Graph. \(y=25-|x|\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.