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Magazine Chapter 2: Problem 20
Graph the line passing through the given point with the given slope. $$ (-3,-1), m=-\frac{1}{5} $$
Short Answer
Expert verified
Graph the line using the equation \( y = -\frac{1}{5}x - \frac{8}{5} \) with points like (-3, -1).
Step by step solution
01
Understand the Slope-Intercept Form
The equation of a line can be expressed in the slope-intercept form as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our task is to find \( b \) using the given information and then graph the line.
02
Substitute the Given Information
We have a point \((-3, -1)\) and a slope \( m = -\frac{1}{5} \). Substitute these into the equation \( y = mx + b \). That gives \( -1 = -\frac{1}{5}(-3) + b \).
03
Solve for the Y-Intercept \( b \)
Calculate \( -\frac{1}{5}(-3) = \frac{3}{5} \). Substitute back to get \( -1 = \frac{3}{5} + b \). To isolate \( b \), subtract \( \frac{3}{5} \) from both sides: \( b = -1 - \frac{3}{5} = -\frac{5}{5} - \frac{3}{5} = -\frac{8}{5} \).
04
Write the Equation of the Line
Now that we have both the slope \( m \) and the y-intercept \( b \), the equation is: \( y = -\frac{1}{5}x - \frac{8}{5} \).
05
Plot the Y-Intercept
On the graph, mark the y-intercept \( b = -\frac{8}{5} \). This point is on the y-axis, approximately \(-1.6\).
06
Use the Slope to Find Another Point
With the slope \( m = -\frac{1}{5} \), for each unit increase in \( x \), \( y \) decreases by \( \frac{1}{5} \). Starting from the y-intercept at \( (-0, -\frac{8}{5}) \), moving over 5 units right gives an exact integer decrement in y by 1: \( (5, -\frac{13}{5}) \) which can be plotted.
07
Draw the Line
Connect the points \((-3, -1)\) and \( (0, -\frac{8}{5}) \) with a straight line using a ruler. Extend this line across the graph to accurately represent the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
Understanding the slope-intercept form is crucial for graphing linear equations because it simplifies the process of finding points on the line. The general formula for the slope-intercept form is given by \( y = mx + b \), where:
- \( m \) represents the slope of the line.
- \( b \) represents the y-intercept, the point where the line crosses the y-axis.
Finding Y-Intercept
To find the y-intercept \( b \) of a line when given a point and the slope, you will substitute the known values into the slope-intercept form \( y = mx + b \). In our example, using the point \((-3, -1)\) and the slope \( m = -\frac{1}{5} \), perform the following:
- Substitute \( x = -3 \) and \( y = -1 \) into the equation, obtaining \( -1 = -\frac{1}{5}(-3) + b \).
- Calculate the expression: \( -\frac{1}{5}(-3) = \frac{3}{5} \).
- Solve for \( b \) by rearranging the equation to \( b = -1 - \frac{3}{5} \).
- Complete the calculation to find \( b = -\frac{8}{5} \).
Plotting Points on a Graph
Plotting points on a graph is the next step after determining the y-intercept and writing the line's equation. Begin by marking the y-intercept, which in our completed equation is \(-\frac{8}{5}\) on the y-axis. This is the starting point for drawing the line. Here are the steps:
- Locate \( b = -\frac{8}{5} \) on the vertical or y-axis.
- Place a point there to signify the y-intercept.
Using Slope to Determine Points
The slope \( m \), which is \(-\frac{1}{5}\), tells us how the line moves across the graph. This means that for every increase of 5 units in the x-direction, the y-value decreases by 1. Here's how to use this information effectively:
- From the y-intercept \((-0, -\frac{8}{5})\), move 5 units to the right on the x-axis.
- Simultaneously, move 1 unit down as indicated by the negative slope, arriving at the new point \((5, -\frac{13}{5})\).
