/*! 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 50 Find the slope and y-intercept o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 50

Find the slope and y-intercept of the line, and draw its graph. $$ -3 x-5 y+30=0 $$

Short Answer

Expert verified
Slope: \(-\frac{3}{5}\); Y-intercept: 6; Graph by plotting (0,6) and using slope.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by manipulating the given equation to match this form. The given equation is \(-3x - 5y + 30 = 0\). To isolate \( y \), first move \(-3x\) and \(30\) to the other side by adding them: \(-5y = 3x - 30\).
02

Solve for y

Continue solving for \( y \) by dividing every term by \(-5\) to get \( y = -\frac{3}{5}x + 6\). This is the slope-intercept form of the line where the coefficient of \( x \) represents the slope \( m \) and the constant term is the y-intercept \( b \).
03

Identify Slope and Y-Intercept

From the slope-intercept form \( y = -\frac{3}{5}x + 6 \), identify that the slope \( m \) is \( -\frac{3}{5} \) and the y-intercept \( b \) is \( 6 \).
04

Graph the Line

Begin by plotting the y-intercept \( (0, 6) \) on the graph. Use the slope \( -\frac{3}{5} \), which means "down 3 units, right 5 units," to find another point. From \( (0, 6) \), move down 3 units and 5 units to the right to reach \( (5, 3) \). Draw a straight line through these points to graph the line.

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.

Y-Intercept
A line's y-intercept is where it crosses the y-axis. This point is pivotal in graphing linear equations, as it provides a starting point to plot the line. In slope-intercept form, expressed as \( y = mx + b \), the \( b \) value is the y-intercept.
To understand further, consider that when \( x = 0 \), the equation simplifies to \( y = b \). This means the line intercepts the y-axis at \( (0, b) \).
In our example equation \( y = -\frac{3}{5}x + 6 \), \( b \) is \( 6 \). Consequently, the y-intercept is at the point \( (0, 6) \).
This point is always plotted first when graphing, helping guide the graph's orientation and position.
Slope
The slope of a line indicates its steepness and direction. Represented by \( m \) in the slope-intercept form \( y = mx + b \), slope reflects how much the y-value changes for a given change in the x-value.
For instance, a slope of \( -\frac{3}{5} \) signifies that for every 5 units you move to the right along the x-axis, you move 3 units down on the y-axis. A positive slope means the line ascends, while a negative slope implies it descends.
  • Zero slope: Line is horizontal.
  • Undefined slope: Line is vertical.
In our example, the line's slope is \( -\frac{3}{5} \). This negative value tells us the line moves downward as it goes from left to right.
Graphing Linear Equations
Graphing linear equations involves plotting a line on a coordinate plane that reflects the equation's slope and y-intercept.
This process starts with transforming the equation into the slope-intercept form \( y = mx + b \). For our line, the equation is \( y = -\frac{3}{5}x + 6 \).
Steps to graphing:
  • First, plot the y-intercept. Here, the intercept is \( (0, 6) \).
  • Use the slope to find another point. For \( -\frac{3}{5} \), start at \( (0, 6) \), move 3 units down and 5 units right to arrive at \( (5, 3) \).
  • Draw a straight line through these points.
This method effectively translates an equation into a visual representation, making it easier to analyze and understand mathematical relationships.

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

Find the slope and y-intercept of the line, and draw its graph. $$ x+3 y=0 $$

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ x^{3}+11 x \leq 6 x^{2}+6 $$

Drug Dosages If the recommended adult dosage for a drug is \(D\) (in mg), then to determine the appropriate dosage \(c\) for a child of age \(a\) , pharmacists use the equation $$ c=0.0417 D(a+1) $$ Suppose the dosage for an adult is 200 \(\mathrm{mg}\) (a) Find the slope. What does it represent? (b) What is the dosage for a newborn?

Find the slope of the line through P and Q. $$ P(2,4), Q(4,3) $$

Collinear Points Suppose you are given the coordinates of three points in the plane and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.