To graph the line passing through the point [tex]\((1, -2)\)[/tex] with a slope of [tex]\(-\frac{2}{3}\)[/tex], let's break down the process step-by-step.
### Step 1: Understand the Slope
The slope [tex]\(-\frac{2}{3}\)[/tex] means that for every 3 units you move to the right (in the positive x-direction), you move 2 units down (in the negative y-direction).
### Step 2: Identify the Given Point
We start with the given point [tex]\((1, -2)\)[/tex] which lies on the line.
### Step 3: Find Two Other Points
We need to calculate two other points that lie on the same line:
#### Point 1: Using [tex]\(x = 0\)[/tex]
- Plug [tex]\(x = 0\)[/tex] into the line equation.
- Given point: [tex]\((1, -2)\)[/tex].
To find the corresponding [tex]\(y\)[/tex]:
[tex]\[ y - (-2) = -\frac{2}{3}(x - 1) \][/tex]
Substitute [tex]\(x = 0\)[/tex]:
[tex]\[ y + 2 = -\frac{2}{3}(0 - 1) \][/tex]
[tex]\[ y + 2 = \frac{2}{3} \][/tex]
[tex]\[ y = \frac{2}{3} - 2 \][/tex]
[tex]\[ y = \frac{2}{3} - \frac{6}{3} \][/tex]
[tex]\[ y = -\frac{4}{3} \][/tex]
So, the first point is [tex]\((0, -\frac{4}{3})\)[/tex].
#### Point 2: Using [tex]\(x = 2\)[/tex]
- Plug [tex]\(x = 2\)[/tex] into the line equation.
- Given point: [tex]\((1, -2)\)[/tex].
To find the corresponding [tex]\(y\)[/tex]:
[tex]\[ y - (-2) = -\frac{2}{3}(x - 1) \][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ y + 2 = -\frac{2}{3}(2 - 1) \][/tex]
[tex]\[ y + 2 = -\frac{2}{3}(1) \][/tex]
[tex]\[ y + 2 = -\frac{2}{3} \][/tex]
[tex]\[ y = -\frac{2}{3} - 2 \][/tex]
[tex]\[ y = -\frac{2}{3} - \frac{6}{3} \][/tex]
[tex]\[ y = -\frac{8}{3} \][/tex]
So, the second point is [tex]\((2, -\frac{8}{3})\)[/tex].
### Summary of Points
- Original Point: [tex]\((1, -2)\)[/tex]
- Point 1: [tex]\((0, -\frac{4}{3})\)[/tex]
- Point 2: [tex]\((2, -\frac{8}{3})\)[/tex]
To provide these points in decimal form:
1. The first point is [tex]\((0, -1.3333333333333335)\)[/tex]
2. The second point is [tex]\((2, -2.6666666666666665)\)[/tex]
### Conclusion
We have found the points:
1. [tex]\((1, -2)\)[/tex]
2. [tex]\((0, -1.3333333333333335)\)[/tex]
3. [tex]\((2, -2.6666666666666665)\)[/tex]
These points lie on the line with a slope of [tex]\(-\frac{2}{3}\)[/tex] passing through the point [tex]\((1, -2)\)[/tex].
