/*! 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 19 line through \((-3,1)\) and \((3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 19

line through \((-3,1)\) and \((3,3)\)

Short Answer

Expert verified
The equation of the line that passes through the points \((-3,1)\) and \((3,3)\) is \(y = \frac{1}{3}x + 2\).

Step by step solution

01

Compute the slope

The slope of a line through points \((x_1,y_1)\) and \((x_2,y_2)\) can be calculated using the formula \[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]. Substituting the values gives us \[m = \frac{{3 - 1}}{{3 - (-3)}} = \frac{2}{6} = \frac{1}{3}\]. So, the slope of the line is \(\frac{1}{3}\).
02

Write the equation

Now that we have the slope, we can use one of the points and the point-slope form \[y - y_1 = m(x - x_1)\] to find the equation of the line. Let's use the point (-3,1). Substituting the values, we get \[y - 1 = \frac{1}{3}(x - (-3))\], simplifying leads to \[y - 1 = \frac{1}{3}x + 1\].
03

Simplify the equation

To get the final equation of the line, isolate y by adding 1 to both sides which gives us \[y = \frac{1}{3}x + 2\]. This is the equation of the line that passes through points \((-3,1)\) and \((3,3)\).

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
Understanding the slope of a line is crucial when studying geometry and algebra. The slope is a measure of how steep a line is. Moreover, it indicates the direction in which the line proceeds, whether upward or downward. To calculate the slope, we use a simple formula:
\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]
where \( m \) represents the slope, and \( (x_1,y_1) \) and \( (x_2,y_2) \) are two different points on the line. In our case, the points are \( (-3,1) \) and \( (3,3) \). By substituting them into the formula, we calculated the slope as \( \frac{1}{3} \), which means for every three units we move horizontally, the vertical movement is one unit up.
Point-Slope Form
Once the slope is determined, the next step is to write the equation of the line. One efficient way to do this is by using the point-slope form of a linear equation:
\[ y - y_1 = m(x - x_1) \]
This form is tremendously helpful since it only requires a single point on the line \( (x_1, y_1) \) and the slope \( m \). In our exercise, we chose the point \( (-3,1) \) and slope \( \frac{1}{3} \) to plug into the formula. As a result, we obtained \( y - 1 = \frac{1}{3}(x - (-3)) \), which ultimately simplified to a neater linear equation. It succinctly represents the unique line passing through the given points.
Linear Equations
Linear equations form the backbone of algebra and have countless applications. They can always be written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis. Referring back to our exercise, after simplifying the point-slope form, we reach the final linear equation \( y = \frac{1}{3}x + 2 \). Here, the slope is \( \frac{1}{3} \) and the y-intercept is 2, making it easy to graph the line. Understanding this equation lets us predict values, graph them, and explore the line's properties that geometrically model our problem-solving.

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

Given points \(A(1,1), B(13,9),\) and \(C(3,7) . D\) is the midpoint of \(\overline{A B},\) and \(E\) is the midpoint of \(\overline{A C}\). a. Find the coordinates of \(D\) and \(E\). b. Use slopes to show that \(\overline{D E} \| \overline{B C}\). c. Use the distance formula to show that \(D E=\frac{1}{2} B C\).

Sketch the graph of \((x-2)^{2}+(y-5)^{2} \leq 9\)

Given points \(A, B,\) and \(C .\) Find \(A B, B C,\) and \(A C .\) Are \(A, B,\) and \(C\) collinear? If so, which point lies between the other two? $$A(3,4), B(-3,0), C(-1,1)$$

Graph the points \(A(-5,0), B(3,2), C(5,6),\) and \(D(-3,4) .\) Then show that \(A B C D\) is a parallelogram by two different methods. a. Show that one pair of opposite sides are both congruent and parallel. b. Show that the diagonals bisect each other (have the same midpoint).

Sketch the graph of \((x-3)^{2}+(y+4)^{2}=36\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure 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.