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Chapter 4: Problem 313

Graph the line of each equation using its slope and \(y\) -intercept. \(y=-\frac{2}{5} x-3\)

Short Answer

Expert verified
Plot points at (0, -3) and (5, -5) then draw a line through them.

Step by step solution

01

Identify the slope and y-intercept

The equation is given in slope-intercept form, which is written as: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. Therefore, the slope \( m \) is \( -\frac{2}{5} \) and the y-intercept \( b \) is \( -3 \).
02

Plot the y-intercept

Locate the y-intercept on the graph. This can be found on the y-axis. The y-intercept is at \( (0, -3) \), so put a point at this coordinate.
03

Apply the slope to find another point

The slope \( -\frac{2}{5} \) means that for every 5 units we move to the right along the x-axis, we move 2 units down along the y-axis. Starting from the y-intercept \( (0, -3) \), move 5 units to the right to \( x = 5 \), and then move 2 units down to \( y = -5 \). This gives another point at \( (5, -5) \).
04

Draw the line

Draw a straight line through the two points \( (0, -3) \) and \( (5, -5) \). This is the graph of the equation \( y = -\frac{2}{5} x - 3 \).

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Key Concepts

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

slope-intercept form
The slope-intercept form is a way to express linear equations. It is written as: \[ y = mx + b \] In this format, \[ m \] stands for the slope of the line, and \[ b \] signifies the y-intercept. This form is beneficial because it directly gives us the slope and y-intercept, making it easier to graph the equation. For example, in the equation \[ y = -\frac{2}{5}x - 3 \], \[ -\frac{2}{5} \] is the slope, and \[ -3 \] is the y-intercept. By using the slope-intercept form, you can quickly identify the necessary components to graph the line accurately.
slope
The slope of a line, denoted by \[ m \], measures the steepness and direction of the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, it is represented as: \[ m = \frac{\Delta y}{\Delta x} \] A positive slope indicates the line rises from left to right, while a negative slope means it falls from left to right. A steeper slope results from larger numerical values. In the equation \[ y = -\frac{2}{5}x - 3 \], the slope \[ m = -\frac{2}{5} \]. This means for every 5 units you move to the right along the x-axis, you must move 2 units down on the y-axis.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. It is represented by \[ b \] in the slope-intercept form \[ y = mx + b \]. At this point, the value of \[ x \] is zero. For the equation \[ y = -\frac{2}{5}x - 3 \], the y-intercept is \[ -3 \]. This tells us that when \[ x \] is zero, \[ y = -3 \]. To graph this, locate the point \[ (0, -3) \] on the y-axis and place a dot there. This dot marks the starting point for graphing your line. By understanding the y-intercept, you can quickly and easily begin plotting the line on a graph.

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Most popular questions from this chapter

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (0,0) and (1,4)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=-x+5,\) point(3,3)

Graph the linear inequality \(y>4 x\)

Graph the linear inequality \(3 x+2 y \geq-6\)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x+5 y=6,\) point(0,0)
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