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To graph the line with a slope of [tex]\(\frac{1}{3}\)[/tex] that passes through the point [tex]\((-4, -5)\)[/tex], you can follow these steps:
### Step 1: Understand the properties of the line
- Point: [tex]\((-4, -5)\)[/tex]
- Slope: [tex]\(\frac{1}{3}\)[/tex]
### Step 2: Write the equation of the line
The equation of a line in point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
For our line:
[tex]\[ y - (-5) = \frac{1}{3}(x - (-4)) \][/tex]
which simplifies to:
[tex]\[ y + 5 = \frac{1}{3}(x + 4) \][/tex]
### Step 3: Solve for [tex]\(y\)[/tex] to get the slope-intercept form [tex]\(y = mx + b\)[/tex]
[tex]\[ y + 5 = \frac{1}{3}(x + 4) \][/tex]
[tex]\[ y + 5 = \frac{1}{3}x + \frac{4}{3} \][/tex]
Subtract 5 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{1}{3}x + \frac{4}{3} - 5 \][/tex]
[tex]\[ y = \frac{1}{3}x + \frac{4}{3} - \frac{15}{3} \][/tex]
[tex]\[ y = \frac{1}{3}x - \frac{11}{3} \][/tex]
Thus, the equation of the line is:
[tex]\[ y = \frac{1}{3}x - \frac{11}{3} \][/tex]
### Step 4: Generate points on the line
Using the equation, we can calculate the [tex]\(y\)[/tex]-values for various [tex]\(x\)[/tex]-values to generate points that lie on the line. Here's a table of some calculated points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -10 & -7 \\ -9.95 & -6.983 \\ -9.90 & -6.967 \\ -9.85 & -6.950 \\ -9.80 & -6.933 \\ -9.75 & -6.916 \\ -9.70 & -6.900 \\ -9.65 & -6.883 \\ -9.60 & -6.866 \\ \vdots & \vdots \\ 10 & 1.889 \\ \hline \end{array} \][/tex]
### Step 5: Plot the points and draw the line
To graph the line, plot the following points:
- At [tex]\( x = -10 \)[/tex], [tex]\( y = -7 \)[/tex]
- At [tex]\( x = -9.95 \)[/tex], [tex]\( y = -6.983 \)[/tex]
- At [tex]\( x = -9.90 \)[/tex], [tex]\( y = -6.967 \)[/tex]
- At [tex]\( x = -9.85 \)[/tex], [tex]\( y = -6.950 \)[/tex]
- At [tex]\( x = -9.80 \)[/tex], [tex]\( y = -6.933 \)[/tex]
- At [tex]\( x = -9.75 \)[/tex], [tex]\( y = -6.916 \)[/tex]
- At [tex]\( x = -9.70 \)[/tex], [tex]\( y = -6.900 \)[/tex]
- At [tex]\( x = -9.65 \)[/tex], [tex]\( y = -6.883 \)[/tex]
- At [tex]\( x = 10 \)[/tex], [tex]\( y = 1.889 \)[/tex]
### Step 6: Connect the points with a straight line
When you connect these points, you'll get a straight line that represents the equation [tex]\( y = \frac{1}{3}x - \frac{11}{3} \)[/tex].
Thus, the line with a slope of [tex]\(\frac{1}{3}\)[/tex] passing through the point [tex]\((-4, -5)\)[/tex] can be drawn as described above and passes through the given points.
### Step 1: Understand the properties of the line
- Point: [tex]\((-4, -5)\)[/tex]
- Slope: [tex]\(\frac{1}{3}\)[/tex]
### Step 2: Write the equation of the line
The equation of a line in point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
For our line:
[tex]\[ y - (-5) = \frac{1}{3}(x - (-4)) \][/tex]
which simplifies to:
[tex]\[ y + 5 = \frac{1}{3}(x + 4) \][/tex]
### Step 3: Solve for [tex]\(y\)[/tex] to get the slope-intercept form [tex]\(y = mx + b\)[/tex]
[tex]\[ y + 5 = \frac{1}{3}(x + 4) \][/tex]
[tex]\[ y + 5 = \frac{1}{3}x + \frac{4}{3} \][/tex]
Subtract 5 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{1}{3}x + \frac{4}{3} - 5 \][/tex]
[tex]\[ y = \frac{1}{3}x + \frac{4}{3} - \frac{15}{3} \][/tex]
[tex]\[ y = \frac{1}{3}x - \frac{11}{3} \][/tex]
Thus, the equation of the line is:
[tex]\[ y = \frac{1}{3}x - \frac{11}{3} \][/tex]
### Step 4: Generate points on the line
Using the equation, we can calculate the [tex]\(y\)[/tex]-values for various [tex]\(x\)[/tex]-values to generate points that lie on the line. Here's a table of some calculated points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -10 & -7 \\ -9.95 & -6.983 \\ -9.90 & -6.967 \\ -9.85 & -6.950 \\ -9.80 & -6.933 \\ -9.75 & -6.916 \\ -9.70 & -6.900 \\ -9.65 & -6.883 \\ -9.60 & -6.866 \\ \vdots & \vdots \\ 10 & 1.889 \\ \hline \end{array} \][/tex]
### Step 5: Plot the points and draw the line
To graph the line, plot the following points:
- At [tex]\( x = -10 \)[/tex], [tex]\( y = -7 \)[/tex]
- At [tex]\( x = -9.95 \)[/tex], [tex]\( y = -6.983 \)[/tex]
- At [tex]\( x = -9.90 \)[/tex], [tex]\( y = -6.967 \)[/tex]
- At [tex]\( x = -9.85 \)[/tex], [tex]\( y = -6.950 \)[/tex]
- At [tex]\( x = -9.80 \)[/tex], [tex]\( y = -6.933 \)[/tex]
- At [tex]\( x = -9.75 \)[/tex], [tex]\( y = -6.916 \)[/tex]
- At [tex]\( x = -9.70 \)[/tex], [tex]\( y = -6.900 \)[/tex]
- At [tex]\( x = -9.65 \)[/tex], [tex]\( y = -6.883 \)[/tex]
- At [tex]\( x = 10 \)[/tex], [tex]\( y = 1.889 \)[/tex]
### Step 6: Connect the points with a straight line
When you connect these points, you'll get a straight line that represents the equation [tex]\( y = \frac{1}{3}x - \frac{11}{3} \)[/tex].
Thus, the line with a slope of [tex]\(\frac{1}{3}\)[/tex] passing through the point [tex]\((-4, -5)\)[/tex] can be drawn as described above and passes through the given points.
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