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Chapter 3: Problem 18

A line with the given slope passes through the given point. Write the equation of the line in slope-intercept form. slope \(=0 ;(8,-1)\)

Short Answer

Expert verified
The equation of the line is y = -1.

Step by step solution

01

Identify given information

The slope of the line is given as 0 and the line passes through the point (8, -1).
02

Understanding 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.
03

Substitute the slope

Given that the slope (m) is 0, substitute m into the slope-intercept form: y = 0x + b This simplifies to y = b
04

Find the y-intercept

The line passes through the point (8, -1). Substitute x = 8 and y = -1 into the equation y = b -1 = b
05

Write the final equation

Combine the results to write the equation of the line: y = -1

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Line
The equation of a line is a mathematical way to express a straight line on a graph. It's written in different forms, but one of the most common is the slope-intercept form: \(y = mx + b\)Where:
  • \(y\) and \(x\) are the coordinates of any point on the line.
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept of the line.
This form allows us to easily see the slope and y-intercept of the line. By plugging in the values for the slope and a point the line passes through, we can determine the equation. Understanding this foundation will help significantly when solving related problems.
Slope
The slope of a line is a measure of its steepness. It describes how much the y-coordinate (vertical position) changes for a change in the x-coordinate (horizontal position). The slope is calculated by the formula: \(m = \frac{{\text{{change in y}}}}{{\text{{change in x}}}} = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)In the given problem, the slope is 0. This means the line is perfectly horizontal, and there is no vertical change regardless of the horizontal movement. Thus, the value of \(y\) remains constant as \(x\) changes. Whenever you encounter a slope of 0, the equation of the line will take the form \(y = b\), where b is the y-intercept.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In slope-intercept form (\(y = mx + b\)), it's represented by \(b\). To find the y-intercept, we look at the point where \(x = 0\). For the given exercise, the line passes through the point (8, -1) with a slope of 0. Using this information, we substitute into the equation and find: \(-1 = b\)Therefore, the y-intercept \(b\) is -1. As a result, the equation of the line in slope-intercept form is: \(y = -1\)This tells us that no matter the value of \(x\), \(y\) will always be -1. So, the line is a horizontal line passing through -1 on the y-axis.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \((-6,3) ;(-9,-2)\)

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=-4\)

(a) find three solutions of the equation. (b) graph the equation. \(y=-x\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-y=8\)

A high-speed Shinkansen train in Japan travels at a speed of \(\frac{270 \mathrm{~km}}{1 \mathrm{hr}}\) for \(18 \mathrm{~min}\). Find the distance it travels.
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