/*! 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 33 Find an equation of the line: w... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 33

Find an equation of the line: with \(y\) -intercept \((0,-6)\) and slope \(\frac{1}{2}\).

Short Answer

Expert verified
The equation of the line is \( y = \frac{1}{2}x - 6 \).

Step by step solution

01

- Understand the slope-intercept form

The equation of a line in slope-intercept form is given by the formula \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept.
02

- Identify the values

From the provided information, the slope (\( m \)) is \( \frac{1}{2} \) and the y-intercept (\( b \)) is -6.
03

- Substitute the values

Substitute the values of the slope and y-intercept into the slope-intercept form equation. This will give \( y = \frac{1}{2}x - 6 \).
04

- Finalize the equation

Write the final equation using the substituted values: \( y = \frac{1}{2}x - 6 \).

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 find the equation of a line, we typically use the **slope-intercept form**. This form is structured as: oin
  • \( y = mx + b \)
Here, \( m \) stands for the **slope** of the line, and \( b \) represents the **y-intercept**. It's a widely used formula because it directly shows where the line crosses the y-axis and how steep the line is. By using this form, you can easily plot the line or understand its behavior visually.
slope
The **slope** of a line indicates how slanted it is. It's a measure of the line's steepness and direction. In mathematical terms, the slope is represented by \( m \). If you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), you can calculate the slope using:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
In our given exercise, the slope (\( m \)) is provided as \( \frac{1}{2} \). This means that for every unit increase in \( x \), the value of \( y \) increases by half a unit. A positive slope like this one tells us that the line rises from left to right.
y-intercept
The **y-intercept** is the point where the line crosses the y-axis. This occurs when \( x = 0 \). In the slope-intercept form equation \( y = mx + b \), the y-intercept corresponds to the value of \( b \). In our exercise, it is given as \( -6 \). This means that the line crosses the y-axis at the point \( (0, -6) \). Knowing the y-intercept helps you plot the starting point of the line on the graph. By understanding both the slope and y-intercept, you can effectively draw and understand the behavior of the line.
substitution in equations
When dealing with equations, **substitution** involves replacing variables with specific values. For our task, we substituted the given slope and y-intercept values into the slope-intercept form equation. Start with:
  • Equation: \( y = mx + b \)
  • Given slope: \( m = \frac{1}{2} \)
  • Given y-intercept: \( b = -6 \)
When we substitute, the equation becomes:
  • \( y = \frac{1}{2}x - 6 \)
Through substitution, we've created the specific equation for our line, confirming the points and slope match precisely. By replacing the values, you see how the line's behavior emerges.

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

Consider the function \(\int\) given by $$f(x)=\left\\{\begin{array}{ll} -2 x+1, & \text { for } x<0 \\ 17, & \text { for } x=0 \\ x^{2}-3, & \text { for } 0

Find the domain of each function given below. $$f(x)=x^{2}+3$$

Red Tide is planning a new line of skis. For the first year, the fixed costs for setting up production are 45,000 dollars. The variable costs for producing each pair of skis are estimated at 80 dollars, and the selling price will be 255 dollars per pair. It is projected that 3000 pairs will sell the first year. a) Find and graph \(C(x),\) the total cost of producing \(x\) pairs of skis. b) Find and graph \(R(x),\) the total revenue from the sale of \(x\) pairs of skis. Use the same axes as in part (a). c) Using the same axes as in part (a), find and graph \(P(x),\) the total profit from the production and sale of \(x\) pairs of skis. d) What profit or loss will the company realize if the expected sale of 3000 pairs occurs? e) How many pairs must the company sell in order to break even?

In computing the dosage for chemotherapy, a patient's body surface area is needed. A good approximation of a person's surface area \(s,\) in square meters \(\left(m^{2}\right),\) is given by the formula $$s=\sqrt{\frac{h w}{3600}},$$ where w is the patient's weight in kilograms (kg) and h is the patient's height in centimeters (cm). (Source: U.S. Oncology.) Use the preceding information. Round your answers to the nearest thousandth. Assume that a patient's weight is 70 kg. Approximate the patient's surface area assuming that: a) The patients height is 150 cm. b) The patients height is 180 cm.

The amount of money, \(A(t),\) in a savings account that pays \(6 \%\) interest, compounded quarterly for \(t\) years, with an initial investment of \(P\) dollars, is given by $$ A(t)=P\left(1+\frac{0.06}{4}\right)^{4 t} $$ If \(\$ 800\) is invested at \(6 \%,\) compounded quarterly, how much will the investment be worth after 3 yr?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.