/*! 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 25 Mixed Practice Find the slope of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 25

Mixed Practice Find the slope of each line. See Examples 3 through 6. $$ 2 x-3 y=10 $$

Short Answer

Expert verified
The slope of the line is \( \frac{2}{3} \).

Step by step solution

01

Rearrange equation to slope-intercept form

The slope-intercept form of a linear equation is given as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We start with the equation \( 2x - 3y = 10 \). We need to rearrange this into the form \( y = mx + b \).
02

Isolate y in the equation

From the equation \( 2x - 3y = 10 \), we want to express \( y \) in terms of \( x \). First, subtract \( 2x \) from both sides to get: \( -3y = -2x + 10 \).
03

Solve for y

Divide every term by \(-3\) to solve for \( y \):\( y = \frac{-2}{-3}x + \frac{10}{-3} \).Simplify to get:\( y = \frac{2}{3}x - \frac{10}{3} \).
04

Identify the slope

From the equation \( y = \frac{2}{3}x - \frac{10}{3} \), it is clear that the slope \( m \) is the coefficient of \( x \), which is \( \frac{2}{3} \).

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-Intercept Form
To understand linear equations, one important form to know is the slope-intercept form. This form is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis, meaning it is the value of \( y \) when \( x \) is zero.
The slope-intercept form is beneficial because it allows us to quickly determine both the steepness of the line (the slope) and where it crosses the vertical axis (the y-intercept). This form is especially useful for graphing linear equations and understanding their behavior in a visual format.
Slope of a Line
The slope of a line is a measure of its steepness and direction. It is usually represented by the letter \( m \) in the slope-intercept form. Mathematically, it is calculated as the change in \( y \) divided by the change in \( x \), often described as "rise over run."
Key characteristics of the slope include:
  • A positive slope indicates that the line rises from left to right.
  • A negative slope means the line falls from left to right.
  • A zero slope describes a horizontal line.
  • An undefined slope (often involving division by zero) describes a vertical line.
Knowing the slope helps us predict how one variable responds to changes in another, making it a fundamental concept in mathematics and data analysis.
Rearranging Equations
Rearranging equations is a skill that involves altering the format of an equation to make another variable the subject. In the context of linear equations, this often means converting a standard form equation into slope-intercept form to easily determine the slope and y-intercept.
For the given equation \( 2x - 3y = 10 \), rearrangement is necessary to write it in the form \( y = mx + b \).
  • Start by moving terms around to isolate terms involving \( y \) on one side.
  • This involves operations like adding, subtracting, or dividing all terms by a certain number.
  • Make sure to perform the same operation on both sides of the equation to maintain equality.
This process is crucial for solving equations and better understanding their graphical interpretation.
Solving for y
Solving for \( y \) means manipulating the equation so that \( y \) stands alone on one side of the equal sign. It's a step that simplifies linear equations, particularly when we want to graph them or determine specific values.
Starting with an equation like \( 2x - 3y = 10 \), the process might look like this:
  • Subtract \( 2x \) from both sides, giving us \( -3y = -2x + 10 \).
  • Divide every term by \( -3 \) to isolate \( y \), resulting in \( y = \frac{2}{3}x - \frac{10}{3} \).
This last form makes it clear what the slope and y-intercept are, helping visualize the line's path on a graph. Understanding this process equips you with the tools to handle a variety of linear equations efficiently.

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 (1,1) is included in each graph. $$ 3 x+4 y<8 $$

Write an inequality whose solutions are all points with coordinates whose sum is at least \(13 .\)

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. The Pool Fun Company has learned that, by pricing a newly released Fun Noodle at \(\$ 3,\) sales will reach 10,000 Fun Noodles per day during the summer. Raising the price to \(\$ 5\) will cause sales to fall to 8000 Fun Noodles per day. a. Assume that the relationship between price and number of Fun Noodles sold is linear and write an equation describing this relationship. Use ordered pairs of the form (price, number sold). b. Predict the daily sales of Fun Noodles if the price is \(\$ 3.50\)

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through (4,7) and (0,0)

Solve. See Example 8. The cost in dollars \(y\) of producing \(x\) computer desks is given by \(y=80 x+5000\). a. Complete the table. $$ \begin{array}{|c|c|c|c|} \hline \boldsymbol{x} & 100 & 200 & 300 \\ \hline \boldsymbol{y} & & & \\ \hline \end{array} $$ b. Find the number of computer desks that can be produced for \(\$ 8600 .\) (Hint: Find \(x\) when \(y=8600 .)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

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.