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Chapter 7: Problem 81

Graph each line passing through the given point and having the given slope $$ (2,-5) ; m=0 $$

Short Answer

Expert verified
y = -5

Step by step solution

01

Understand the Slope

The slope given is 0 (m=0). A slope of 0 means the line is horizontal.
02

Identify the Point

The given point is (2, -5). This point will be used to find the equation of the line.
03

Write the Equation

For a horizontal line passing through a point (x, y) with slope 0, the equation is always y = constant. Here, the y-coordinate of the given point is -5. Thus, the equation of the line is y = -5.
04

Graph the Line

To graph the line y = -5, simply draw a horizontal line that crosses the y-axis at -5.

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Key Concepts

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

Slope
The slope of a line measures its steepness and direction. Mathematically, the slope (m) is defined as the change in the y-coordinate (rise) divided by the change in the x-coordinate (run) between two points on the line. When the slope is 0, it means the line is perfectly horizontal. This indicates that there is no change in the y-coordinate, no matter how much the x-coordinate changes. In other words, the line is flat, and it runs parallel to the x-axis. Understanding the slope is crucial when you're graphing linear equations because it tells you how to orient your line on the graph.
Horizontal Line
A horizontal line runs parallel to the x-axis and has a slope of 0. This means there is no vertical change as you move along the line. The y-coordinate remains constant for all points on the line. For instance, if you have a horizontal line at y = -5, this means every point on that line has a y-coordinate of -5. To plot a horizontal line on a graph, you find the y-coordinate on the y-axis and draw a straight line parallel to the x-axis across the graph. This simple approach makes horizontal lines easy to identify and graph.
Equation of a Line
The equation of a line expresses the relationship between the x and y coordinates of any point on the line. For a line with a slope of 0, this relationship simplifies significantly because the line is horizontal. The general form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. However, for horizontal lines, since m = 0, the equation simplifies to y = b. In the given exercise, the point (2, -5) indicates that the y-coordinate is always -5, resulting in the equation y = -5. This equation tells us that every point on this line has a y-coordinate of -5, regardless of the x-coordinate.

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Most popular questions from this chapter

Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} 3 x=27+2 y \\ x-\frac{7}{2} y=-25 \end{array} $$

Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(5 x+4 y=7\) \(10 x+8 y=4\)

Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(x-3 y=5\) \(2 x+y=8\)

Solve each system. If the system is inconsistent or has dependent equations, say so. $$ \begin{array}{l} 4 x+y-2 z=3 \\ x+\frac{1}{4} y-\frac{1}{2} z=\frac{3}{4} \\ 2 x+\frac{1}{2} y-\quad z=1 \end{array} $$

Determine whether each pair of lines is parallel, perpendicular, or neither $$ x+1=0 \text { and } y-2=0 $$
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