/*! 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 7 Find an equation for the line th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

Find an equation for the line that passes through the given points. $$ (-2,1) \text { and }(2,3) $$

Short Answer

Expert verified
The equation of the line is \(y = \frac{1}{2}x + 2\).

Step by step solution

01

Determine the Slope

To find the slope of the line that passes through the points \((-2, 1)\) and \((2, 3)\), we use the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the given points, \(y_2 = 3\), \(y_1 = 1\), \(x_2 = 2\), and \(x_1 = -2\):\[m = \frac{3 - 1}{2 - (-2)} = \frac{2}{4} = \frac{1}{2}\]Thus, the slope \(m\) is \(\frac{1}{2}\).
02

Use Point-Slope Form

With the slope \(m = \frac{1}{2}\) and one of the points \((-2, 1)\), we apply the point-slope form of a line: \[y - y_1 = m(x - x_1)\]Substitute \(m = \frac{1}{2}\), \(x_1 = -2\), and \(y_1 = 1\):\[y - 1 = \frac{1}{2}(x - (-2))\]Simplify the equation:\[y - 1 = \frac{1}{2}(x + 2)\]
03

Convert to Slope-Intercept Form

To express the equation in the slope-intercept form \(y = mx + b\), simplify the equation from Step 2:\[y - 1 = \frac{1}{2}(x + 2)\]Distribute the \(\frac{1}{2}\):\[y - 1 = \frac{1}{2}x + 1\]Add 1 to both sides to solve for \(y\):\[y = \frac{1}{2}x + 2\]

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 find the equation of a line, you first need to determine its slope. The slope of a line measures its steepness and can be calculated using two points on the line. These are typically labeled \(x_1, y_1\) and \(x_2, y_2\). The formula for slope, usually denoted by \(m\), is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In simple words, you subtract the y-coordinates of the points to find the rise, and subtract the x-coordinates to find the run. Divide these values, and you鈥檒l get the slope.
  • Let's substitute the points \((-2, 1)\) and \(2, 3)\).
  • Substitute into the formula: \(y_2 = 3\), \(y_1 = 1\), \(x_2 = 2\), and \(x_1 = -2\).
  • Perform the calculations: \[m = \frac{3 - 1}{2 - (-2)} = \frac{2}{4} = \frac{1}{2}\]
Therefore, the slope of the line passing through these points is \(\frac{1}{2}\).
Point-Slope Form
Once the slope \(m\) is known, you can use the point-slope form to write the equation of the line. This is especially useful when you know the slope and one point on the line.
The point-slope form is given by:
\[y - y_1 = m(x - x_1)\]
Let's plug in the slope \(\frac{1}{2}\) and one of the points \((-2, 1)\):
  • Substitute \(m = \frac{1}{2}\), \(x_1 = -2\), and \(y_1 = 1\):
  • \[y - 1 = \frac{1}{2}(x - (-2))\]
  • Simplifying the equation: \[y - 1 = \frac{1}{2}(x + 2)\]
This formula helps you see how the line operates around the point you already have, and it can be easily transformed into other forms like the slope-intercept form.
Slope-Intercept Form
The slope-intercept form is perhaps the most commonly used form of a linear equation because it directly shows the slope and the y-intercept.
This form is expressed as:
\[y = mx + b\]
Where \(m\) is the slope and \(b\) is the y-intercept. Starting from the simplified point-slope form, you can convert it into this form.
  • Start with: \[y - 1 = \frac{1}{2}(x + 2)\]
  • Distribute the fraction: \[y - 1 = \frac{1}{2}x + 1\]
  • To solve for \(y\), add 1 to both sides: \[y = \frac{1}{2}x + 2\]
Now, the equation is in slope-intercept form, and you can clearly see the slope is \(\frac{1}{2}\) and the line crosses the y-axis at 2. This makes it easier to graph and understand the line's characteristics.

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

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{x}{5}$$

You want to invest money for your child's education in a certificate of deposit (CD). You want it to be worth $$\$ 12,000$$ in 10 years. How much should you invest if the CD pays interest at a \(9 \%\) annual rate compounded (a) Annually? (b) Continuously?

Write a formula representing the function. The gravitational force, \(F\), between two bodies is inversely proportional to the square of the distance \(d\) between them.

Find the doubling time of a quantity that is increasing by \(7 \%\) per year.

Sketch graphs of the functions. What are their amplitudes and periods? $$y=-3 \sin 2 \theta$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.