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Chapter 3: Problem 46

Graph each line passing through the given point and having the given slope. $$ (3,2), m=0 $$

Short Answer

Expert verified
The equation of the line is \(y = 2\).

Step by step solution

01

Identify the Point and Slope

The problem provides the point \(3, 2\) and the slope \(m = 0\). A slope of zero means the line is horizontal.
02

Write the Equation of the Line

For a horizontal line passing through the point \(3,2\), the equation is \(y = 2\) because the y-value does not change.
03

Plot the Given Point

On a graph, plot the given point \(3, 2\). This will be one of the points through which the line passes.
04

Draw the Horizontal Line

From the point \(3, 2\), draw a horizontal line that extends left and right. Since the slope is zero, the line will remain at the same y-value (2) across all x-values.

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Key Concepts

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

Understanding Slope
The slope (often denoted as 'm') of a line represents its steepness and direction. It is calculated as the 'rise' over the 'run', or the vertical change over the horizontal change between two points on the line. For instance, if a line passes through points \(x_1, y_1\) and \(x_2, y_2\), then the slope is computed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. A slope of zero (\(m=0\)) means the line is perfectly horizontal, with no vertical change regardless of how far you move horizontally.
The Characteristics of a Horizontal Line
A horizontal line has a unique feature: it runs parallel to the x-axis and every point on this line has the same y-coordinate. For a line to be horizontal, the slope must be zero (\(m = 0\)). This means there is no vertical change as you move along the line; it simply stays flat. For example, the equation \[y = 2\] represents a horizontal line where every point on the line has a y-value of 2.
Plotting Points on a Graph
Plotting points is the foundation of graphing equations. Each point on a graph is defined by an \(x\) and \(y\) coordinate. To plot a point \((x, y)\), move 'x' units along the x-axis and 'y' units along the y-axis. For example, for the point \(3, 2\), you move 3 units to the right (positive x-direction) and 2 units up (positive y-direction). Mark this point on the graph.
Introduction to Linear Equations
Linear equations describe straight lines on a graph. They can be written in several forms, the most common being slope-intercept form: \[ y = mx + b \], where \(m\) is the slope and \(b\) is the y-intercept (where the line crosses the y-axis). In this exercise, the equation is \[ y = 2 \], a specific case of a linear equation that denotes a horizontal line. Linear equations can also be written in point-slope form: \[ y - y_1 = m(x - x_1) \], emphasizing the line's slope and a specific point it passes through.

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Most popular questions from this chapter

Write an equation of the line satisfying the given conditions. Give the final answer in slope intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines in Section \(3.3 .)\) Perpendicular to \(x-2 y=7 ; \quad y\) -intercept \((0,-3)\)

Concept Check In Exercises \(55-62,\) describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 5 y=-15 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines in Section \(3.3 .)\) Through \((2,-3) ;\) parallel to \(3 x=4 y+5\)

A point \((x, y)\) has the property that \(x y>0 .\) In which quadrant(s) must the point lie? Explain.

Write an equation for each line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. $$ (-1,-7) \text { and }(-8,-2) $$
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