Answer :
To graph a line that contains the point [tex]\((-5, -6)\)[/tex] and has a slope of [tex]\(\frac{2}{3}\)[/tex], follow these steps:
### Understanding the Point-Slope Form
The point-slope form of a linear equation is given by:
[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 of the line.
### Step-by-Step Solution:
1. Identify the given information:
- Point: [tex]\((-5, -6)\)[/tex]
- Slope: [tex]\(\frac{2}{3}\)[/tex]
2. Write the point-slope form equation:
Using the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex], substitute [tex]\((x_1, y_1)\)[/tex] with [tex]\((-5, -6)\)[/tex] and [tex]\(m\)[/tex] with [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ y - (-6) = \frac{2}{3}(x - (-5)) \][/tex]
Simplify:
[tex]\[ y + 6 = \frac{2}{3}(x + 5) \][/tex]
3. Distribute the slope:
[tex]\[ y + 6 = \frac{2}{3}x + \frac{2}{3} \cdot 5 \][/tex]
[tex]\[ y + 6 = \frac{2}{3}x + \frac{10}{3} \][/tex]
4. Isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{2}{3}x + \frac{10}{3} - 6 \][/tex]
To combine the constant terms, express [tex]\(-6\)[/tex] with a denominator of 3:
[tex]\[ y = \frac{2}{3}x + \frac{10}{3} - \frac{18}{3} \][/tex]
[tex]\[ y = \frac{2}{3}x - \frac{8}{3} \][/tex]
This is the slope-intercept form [tex]\(y = mx + b\)[/tex] of the line.
5. Plot the line using the equation [tex]\(y = \frac{2}{3}x - \frac{8}{3}\)[/tex]:
- First, plot the given point [tex]\((-5, -6)\)[/tex] on the graph.
- Use the slope [tex]\(\frac{2}{3}\)[/tex] to find another point. The slope [tex]\(\frac{2}{3}\)[/tex] means that for every 3 units you move horizontally to the right, you move 2 units up.
- Start at [tex]\((-5, -6)\)[/tex]:
1. Move 3 units to the right:
[tex]\(-5 + 3 = -2\)[/tex]
2. Move 2 units up:
[tex]\(-6 + 2 = -4\)[/tex]
This gives another point [tex]\((-2, -4)\)[/tex].
6. Draw the line:
- Plot the points [tex]\((-5, -6)\)[/tex] and [tex]\((-2, -4)\)[/tex].
- Draw a straight line through these points extending in both directions.
The plotted graph is the required line passing through the point [tex]\((-5, -6)\)[/tex] with a slope of [tex]\(\frac{2}{3}\)[/tex].
### Visual Representation
The procedure described will yield a straight line on the coordinate plane. Here's what it will look like:
```
y
↑
|
-4 | (-2,-4)
|
-6 | (-5,-6)
|
+------------------→ x
-5 -4 -3 -2
```
The stars (*) represent the points [tex]\((-5, -6)\)[/tex] and [tex]\((-2, -4)\)[/tex]. Drawing a line through these points will give you the required graph.
### Understanding the Point-Slope Form
The point-slope form of a linear equation is given by:
[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 of the line.
### Step-by-Step Solution:
1. Identify the given information:
- Point: [tex]\((-5, -6)\)[/tex]
- Slope: [tex]\(\frac{2}{3}\)[/tex]
2. Write the point-slope form equation:
Using the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex], substitute [tex]\((x_1, y_1)\)[/tex] with [tex]\((-5, -6)\)[/tex] and [tex]\(m\)[/tex] with [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ y - (-6) = \frac{2}{3}(x - (-5)) \][/tex]
Simplify:
[tex]\[ y + 6 = \frac{2}{3}(x + 5) \][/tex]
3. Distribute the slope:
[tex]\[ y + 6 = \frac{2}{3}x + \frac{2}{3} \cdot 5 \][/tex]
[tex]\[ y + 6 = \frac{2}{3}x + \frac{10}{3} \][/tex]
4. Isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{2}{3}x + \frac{10}{3} - 6 \][/tex]
To combine the constant terms, express [tex]\(-6\)[/tex] with a denominator of 3:
[tex]\[ y = \frac{2}{3}x + \frac{10}{3} - \frac{18}{3} \][/tex]
[tex]\[ y = \frac{2}{3}x - \frac{8}{3} \][/tex]
This is the slope-intercept form [tex]\(y = mx + b\)[/tex] of the line.
5. Plot the line using the equation [tex]\(y = \frac{2}{3}x - \frac{8}{3}\)[/tex]:
- First, plot the given point [tex]\((-5, -6)\)[/tex] on the graph.
- Use the slope [tex]\(\frac{2}{3}\)[/tex] to find another point. The slope [tex]\(\frac{2}{3}\)[/tex] means that for every 3 units you move horizontally to the right, you move 2 units up.
- Start at [tex]\((-5, -6)\)[/tex]:
1. Move 3 units to the right:
[tex]\(-5 + 3 = -2\)[/tex]
2. Move 2 units up:
[tex]\(-6 + 2 = -4\)[/tex]
This gives another point [tex]\((-2, -4)\)[/tex].
6. Draw the line:
- Plot the points [tex]\((-5, -6)\)[/tex] and [tex]\((-2, -4)\)[/tex].
- Draw a straight line through these points extending in both directions.
The plotted graph is the required line passing through the point [tex]\((-5, -6)\)[/tex] with a slope of [tex]\(\frac{2}{3}\)[/tex].
### Visual Representation
The procedure described will yield a straight line on the coordinate plane. Here's what it will look like:
```
y
↑
|
-4 | (-2,-4)
|
-6 | (-5,-6)
|
+------------------→ x
-5 -4 -3 -2
```
The stars (*) represent the points [tex]\((-5, -6)\)[/tex] and [tex]\((-2, -4)\)[/tex]. Drawing a line through these points will give you the required graph.
