/*! 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 56 Find an equation of the line pas... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 56

Find an equation of the line passing through the given points. $$(1,4),(3,4)$$

Short Answer

Expert verified
The equation of the line passing through the points (1,4) and (3,4) is \(y = 4\).

Step by step solution

01

Calculate the Slope

The slope (m) is calculated by the formula: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)\nIn this case, (1,4) is (x1,y1) and (3,4) is (x2,y2).\nSo, \(m = \frac{{4 - 4}}{{3 - 1}} = 0.\)
02

Substituting the Slope and the Point into the Equation

The standard equation for a line is \(y = mx + b\), where m stands for the slope and b for the y-intercept. Substitute the value of m and the coordinates of one point into the equation. Taking the point (1,4): \(4 = 0*1 + b.\) Solving for b gives \(b = 4\).
03

Writing the Equation of the Line

By substituting the values of slope (m) and y-intercept (b) into the equation, the equation of the line becomes \(y = 0x + 4\) or simply \(y = 4\).

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 Calculation
When trying to find the slope of a line, you are determining how steep the line is. It's like asking how much the line rises or falls as it moves horizontally. The formula for slope \(m\) is: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). This formula calculates the change in the \(y\)-coordinates (vertical change) over the change in the \(x\)-coordinates (horizontal change). Let's break it down:
  • \(y_2\) and \(y_1\) represent two \(y\)-values from the different points on the line.
  • \(x_2\) and \(x_1\) are the corresponding \(x\)-values for these points.
For the points \((1, 4)\) and \((3, 4)\), the slope calculation would be: \[m = \frac{{4 - 4}}{{3 - 1}} = \frac{0}{2} = 0\] This result of zero means the line is perfectly horizontal, indicating there is no vertical change as you move along the line horizontally.
Y-intercept
The y-intercept is a crucial feature of a line. It tells us where the line crosses the \(y\)-axis. This is the point at which \(x = 0\). In the equation \(y = mx + b\), \(b\) stands for the y-intercept. To find it, you plug in the slope \(m\) and the coordinates of a point on the line into the equation and solve for \(b\). Let's apply this to our calculated slope and one of the points:
  • Substitute \(m = 0\) and the point \((1, 4)\) into \(y = mx + b\).
  • The setup is: \(4 = 0 \times 1 + b\).
  • Solving gives \(b = 4\).
This \(b\) value of 4 tells us that the line crosses the \(y\)-axis at the point \((0, 4)\). This means no matter how much \(x\) changes, \(y\) remains 4.
Equation of a Line
Finally, we're ready to write the equation of the line. When given the slope \(m\) and the y-intercept \(b\), it's easy to plug these into the format \(y = mx + b\). This equation shows the relationship between any \(x\) and \(y\) on the line.
  • In our example, with \(m = 0\) and \(b = 4\), the line equation becomes: \(y = 0x + 4\).
  • Since \(0x\) equates to zero, the simplified equation is \(y = 4\).
This indicates that for all values of \(x\), \(y\) is always 4. Essentially, it describes a horizontal line extended indefinitely to the left and right across the \(y = 4\) level. In mathematical terms, this is a constant function since it keeps \(y\) value unchanged throughout its domain.

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 adjoint of the matrix \(A .\) Then use the adjoint to find the inverse of \(A,\) if possible. $$A=\left[\begin{array}{rrr} 0 & 1 & 1 \\ 1 & 2 & 3 \\ -1 & -1 & -2 \end{array}\right]$$

Use Cramer's Rule to solve the system of linear equations, if possible. $$\begin{array}{l} 3 x_{1}+4 x_{2}+4 x_{3}=11 \\ 4 x_{1}-4 x_{2}+6 x_{3}=11 \\ 6 x_{1}-6 x_{2}=3 \end{array}$$

The table below shows the projected values (in millions of dollars) of hardback college textbooks sold in the United States for the years 2007 to \(2009 .\) (Source: U.S. Census Bureau) $$\begin{array}{l|c} \hline \text {Year} & \text {Value} \\ \hline 2007 & 4380 \\ 2008 & 4439 \\ 2009 & 4524 \\ \hline \end{array}$$ (a) Create a system of linear equations for the data to fit the curve \(y=a t^{2}+b t+c,\) where \(t\) is the year and \(t=7\) corresponds to \(2007,\) and \(y\) is the value of the textbooks. (b) Use Cramer's Rule to solve your system. (c) Use a graphing utility to plot the data and graph your regression polynomial function. (d) Briefly describe how well the polynomial function fits the data.

Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{aligned} 3 x_{1}-2 x_{2}+9 x_{3}+4 x_{4} &=35 \\ -x_{1} \quad-9 x_{3}-6 x_{4}=-17 \\ 2 x_{3}+x_{4} =5 \\ 2 x_{1}+2 x_{2}\quad\quad +8 x_{4}=-4 \end{aligned}$$

Find the area of the triangle having the given vertices. $$(-1,2),(2,2),(-2,4)$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

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.