/*! 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 15 Find an equation for the line wi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 15

Find an equation for the line with slope \(m\) through the point \((a, c)\)

Short Answer

Expert verified
The equation is \( y = mx - ma + c \).

Step by step solution

01

Understand 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.
02

Substitute Slope and Point into the Equation

Use the point \(a, c\) and the slope \ m \ to find the y-intercept \ b \:\[ c = ma + b \]
03

Solve for y-intercept \(b\)

Rearrange the equation from Step 2 to solve for \(b\):\[ b = c - ma \]
04

Write the Equation of the Line

Now substitute the values of \(m\) and \(b\) back into the slope-intercept form:\[ y = mx + (c - ma) \]
05

Simplify the Final Equation

The final equation for the line is:\[ y = mx - ma + c \]

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 is one of the most familiar ways to express the equation of a line. It's written as \(y = mx + b\). Here, \(m\) stands for the slope of the line, which indicates its steepness and direction. Meanwhile, \(b\) is the y-intercept, the point where the line crosses the y-axis. To put it simply:
  • \(m\) is your runway, guiding how steeply your line will swoop up or down.
  • \(b\) is your starting point, showing where on the y-axis you begin your journey.
By following this form, you can easily sketch out lines and understand their behaviors just by looking at numbers!
Y-Intercept
The y-intercept is the point where your line intersects with the y-axis. This means it's the value of \(y\) when \(x\) is zero. In the slope-intercept form equation, \(y = mx + b\), the \(b\) represents this special y-intercept value. Imagine the y-intercept as the first stop on a bus route that starts at the y-axis. This is where every journey on this line kicks off. You can figure it out easily by plugging in other known values into the slope-intercept equation and solving for \(b\). So, if you know a point \((a, c)\) on the line and the line's slope \(m\), just substitute these into the equation \(c = ma + b\) and solve for \(b\) to find your y-intercept!
Slope
The slope determines how slanted the line is on a graph. It's an incredibly handy concept in algebra to describe how a line moves. The slope, indicated by \(m\) in the equation \(y = mx + b\), calculates as the "rise over run." This means it's the change in \(y\) divided by the change in \(x\) between two points. To think of math visually:
  • If \(m\) is positive, the line rises as it moves to the right.
  • If \(m\) is negative, the line falls as it moves to the right.
  • If \(m\) is zero, the line is flat and horizontal.
  • If \(m\) is undefined, the line is vertical.
So, in one line of an equation, the slope tells you precisely how the line dances across your graph.

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

For the given constant \(c\) and function \(f(x),\) find a function \(g(x)\) that has a hole in its graph at \(x=c\) but \(f(x)=g(x)\) everywhere else that \(f(x)\) is defined. Give the coordinates of the hole. $$f(x)=\ln x, c=1$$

Are the statements true or false? Explain. $$\text { If } \lim _{x \rightarrow c^{+}} g(x)=1 \text { and } \lim _{x \rightarrow c^{-}} g(x)=-1 \text { and } \lim _{x \rightarrow c} \frac{f(x)}{g(x)} \mathrm{exists, then } \lim _{x \rightarrow c} f(x)=0$$

Suppose that \(\lim _{x \rightarrow 3} f(x)=7 .\) Are the statements true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample. If \(\lim _{x \rightarrow 3}(f(x)+g(x))=12,\) then \(\lim _{x \rightarrow 3} g(x)=5\).

At time \(t\) hours after taking the cough suppressant hydrocodone bitartrate, the amount, \(A,\) in mg, remaining in the body is given by \(A=10(0.82)^{t}.\) (a) What was the initial amount taken? (b) What percent of the drug leaves the body each hour? (c) How much of the drug is left in the body 6 hours after the dose is administered? (d) How long is it until only 1 mg of the drug remains in the body?

Give an explanation for your answer. The graph of \(g(x)=\log (x-1)\) crosses the \(x\) -axis at \(x=1.\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

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.