/*! 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 68 Find an equation of the line tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 68

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((4,1) ;\) parallel to \(2 x+5 y=10\)

Short Answer

Expert verified
(a) Slope-intercept form: \( y = -\frac{2}{5}x + \frac{13}{5} \)(b) Standard form: \( 2x + 5y = 13 \)

Step by step solution

01

- Identify the slope of the given line

Rewrite the given equation in slope-intercept form to identify the slope. The equation given is \(2x + 5y = 10\). To convert this to slope-intercept form, solve for \(y\). \[ 2x + 5y = 10 \] Subtract \(2x\) from both sides: \[ 5y = -2x + 10 \] Divide everything by 5: \[ y = -\frac{2}{5}x + 2 \] The slope of the line is \(-\frac{2}{5}\).
02

- Use the slope and point-slope form to find the equation

The slope of the line that is parallel will be the same, so \(m = -\frac{2}{5}\). Now use the point-slope form of the equation \(y - y_1 = m(x - x_1)\) with the given point (4, 1). \[ y - 1 = -\frac{2}{5}(x - 4) \] Simplify this equation to get slope-intercept form.
03

- Convert to slope-intercept form

Distribute \(-\frac{2}{5}\) in the equation: \[ y - 1 = -\frac{2}{5}x + \frac{8}{5} \] Add 1 to both sides to solve for \(y\): \[ y = -\frac{2}{5}x + \frac{8}{5} + 1 \] Combine like terms: \[ y = -\frac{2}{5}x + \frac{8}{5} + \frac{5}{5} \] \[ y = -\frac{2}{5}x + \frac{13}{5} \] So the slope-intercept form is \( y = -\frac{2}{5}x + \frac{13}{5} \).
04

- Convert to standard form

To convert the slope-intercept form \( y = -\frac{2}{5}x + \frac{13}{5} \) to standard form \(Ax + By = C\), first multiply everything by 5 to clear the fractions: \[ 5y = -2x + 13 \] Rearrange terms to get the standard form: \[ 2x + 5y = 13 \] So the standard form is \( 2x + 5y = 13 \).

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-intercept form
The slope-intercept form of a line's equation is one of the most common and useful ways to represent a linear equation. Its general format is \( y = mx + b \), where:
  • \

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

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Use the slope formula to determine whether the points \((1,-2),(3,-1),\) and \((5,0)\) are collinear.

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 4 y=3 x $$

Solve each inequality. $$ -x \leq 0 $$

For each function, find \((a) f(2)\) and \((b) f(-1) .\) See Examples 4 and \(5 .\) $$ f=\\{(2,5),(3,9),(-1,11),(5,3)\\} $$

Find the midpoint of each segment with the given endpoints. $$ (4,-3) \text { and }(-1,3) $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.