/*! 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 2 Rewrite each equation in interce... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 2

Rewrite each equation in intercept form. Show your steps. Check your answer by using a calculator graph or table. a. \(y=14+3(x-5)\) b. \(y=-5-2(x+5)\) (a) c. \(6 x+2 y=24\)

Short Answer

Expert verified
a: y = 3x - 1, b: y = -2x - 15, c: y = -3x + 12.

Step by step solution

01

Simplify the Equation for Part (a)

The original equation is given as \(y = 14 + 3(x-5)\). Start by applying the distributive property to the term \(3(x-5)\). This gives us \(y = 14 + 3x - 15\). Now, simplify the equation by combining like terms: \(y = 3x - 1\). This is now in the form \(y = mx + b\), where the y-intercept \(b = -1\).
02

Simplify the Equation for Part (b)

The original equation is \(y = -5 -2(x+5)\). Apply the distributive property to \(-2(x+5)\), resulting in \(y = -5 - 2x - 10\). Combine like terms to simplify: \(y = -2x - 15\). This is in the form \(y = mx + b\), where the y-intercept \(b = -15\).
03

Rewrite Equation for Part (c) in Slope-Intercept Form

The original equation given is \(6x + 2y = 24\). To find the y-intercept form (\(y = mx + b\)), first, solve for \(y\). Subtract \(6x\) from both sides: \(2y = -6x + 24\). Then, divide every term by 2: \(y = -3x + 12\). This is the y-intercept form and indicates the y-intercept \(b = 12\).
04

Verify Each Result

For verification, input each equation into a graphing calculator or plot them to make sure they cross the y-axis at their calculated intercepts: For part (a), verify that \(y = 3x - 1\) crosses the y-axis at \(-1\). For part (b), confirm \(y = -2x - 15\) intersects at \(-15\). For part (c), check \(y = -3x + 12\) crosses at \(12\).

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 linear equation is a way to express the equation easily showing two important properties about the line: the slope and the y-intercept. It is typically written as \(y = mx + b\), where:
  • \(m\) represents the slope of the line, which tells us how steep the line is. It represents the change in \(y\) for a unit change in \(x\).
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
Understanding this form makes it easier to graph the line and quickly understand its behavior as the slope and intercept are evident at a glance. Many equations can be rearranged into this form by solving for \(y\). This standard form helps in comparing different linear equations and understanding the graph's intersection points and inclinations.
Distributive Property
The distributive property is a useful algebraic tool that helps simplify expressions and equations, particularly when you're dealing with expressions inside parentheses. It is expressed as \(a(b + c) = ab + ac\). Let's break this down:
  • If you have a number outside the parentheses, you "distribute" it by multiplying it with each term inside the parentheses.
  • This method helps to simplify the equation to a form where like terms can be combined.
For example, in the original equation from Step 1, \(3(x-5)\) becomes \(3x - 15\) using the distributive property, effectively removing the parentheses and making the equation easier to work with.
Solving Equations
Solving equations involves finding a value for the variable that makes the equation true. In terms of linear equations, this usually means converting an equation to a form where the solutions (values of \(x\) and \(y\)) are clear, often involving rearrangement.Here's a general approach:
  • Use properties like the distributive property to expand the equation if necessary.
  • Combine like terms, which may involve adding or subtracting terms from both sides of the equation.
  • If the equation is not already solved for \(y\), isolate \(y\) by adding, subtracting, multiplying, or dividing both sides of the equation by necessary numbers or variables.
This step-by-step approach was exemplified in Step 3 of the original solution, where the equation \(6x + 2y = 24\) was transformed by isolating \(y\) to make it \(y = -3x + 12\), a clear slope-intercept form.
Y-Intercept
The y-intercept of a line is the value of \(y\) where the line crosses the y-axis. In the equation \(y = mx + b\), the y-intercept is the constant \(b\).Understanding the y-intercept is important because:
  • It provides a starting point for graphing the line.
  • It remains constant for any line, regardless of changes in the \(x\) variable.
When checking a line graphically, you can confirm if the y-intercept is correct by observing where the line meets the y-axis.In the solutions provided, each part of the exercise concluded by identifying the y-intercept derived after rearranging the equation:- For instance, in Part (a), the y-intercept was found to be \(-1\), indicating where the line \(y = 3x - 1\) crosses the y-axis.

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

An equation of a line is \(y=25-2(x+5)\). a. Name the point used to write the point-slope equation. (Ti) b. Find \(x\) when \(y\) is 15 .

Given the slope of a line and one point on the line, name two other points on the same line. Then use the slope formula to check that the slope between each of the two new points and the given point is the same as the given slope. a. Slope \(\frac{3}{1}\); point \((0,4)\) (a) b. Slope \(-5\); point \((2,8)\) c. Slope \(-\frac{3}{4}\); point \((8,6)\) d. Slope \(0.2\); point \((5,7)\)

Use the distributive property to rewrite each expression without parentheses. a. \(3(x-2)\) b. \(-4(x-5)\) c. \(-2(x+8)\)

Moe Beel has a new cell phone service that is billed at a base fee of \(\mathrm{S} 15\) per month, plus 45 e for each minute the phone is used. Consider the relationship between the time the phone is used and the total monthly cost. Let \(x\) represent time, in minutes, and let \(y\) represent cost, in dollars. a. Give one point on the line, and state the slope of the line in dollars per minute. (A) b. Write the equation of the line. Sketch its graph for the first 30 minutes. c. How will the graph change if Moe adds Call Forwarding, changing the base fee to \(\$ 20\) ? d. How will the graph change if Moe drops Caller ID and Voice Mail so that there is no monthly base fee? e. How will the graph change if instead Moe adds the Text Messaging option, increasing his rate to 55 e per minute?

Write each equation in the form requested. Check your answers by graphing on your calculator. a. Write \(y=13.6(x-1902)+158.2\) in intercept form. b. Write \(y=-5.2 x+15\) in point-slope form using \(x=10\) as the first coordinate of the point.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.