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Sagot :
### Step-by-Step Solution:
To graph the equation [tex]\( y = -\frac{2}{3}x + 5 \)[/tex], we will utilize the slope and the y-intercept provided in the equation. Here is a detailed step-by-step solution:
#### Step 1: Plot the Ordered Pair that Represents the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
In our equation, the y-intercept is 5. Therefore, the ordered pair for the y-intercept is:
[tex]\[ (0, 5) \][/tex]
### Step 2: Use the Slope of the Line to Find Another Ordered Pair That Lies on the Line
The slope of a line indicates the change in y for a given change in x. In this equation, the slope is [tex]\(-\frac{2}{3}\)[/tex]. This means that for every increase of 3 units in the x-direction (positive x-direction), the y-value decreases by 2 units (negative y-direction).
Starting from the y-intercept point [tex]\((0, 5)\)[/tex]:
1. Move 3 units to the right along the x-axis:
[tex]\[ x = 0 + 3 = 3 \][/tex]
2. Move 2 units down along the y-axis because the slope is negative:
[tex]\[ y = 5 - 2 = 3 \][/tex]
However, the correct calculation provides another moving direction. Let's verify:
Given the slope [tex]\( -\frac{2}{3} \)[/tex]:
- When moving 3 units to the right, in terms of x, we get the new x-coordinate as 3.
- The corresponding change in y is calculated using the slope. The decrease in y ([tex]\(\Delta y\)[/tex]) is:
[tex]\[ \Delta y = -\left(\frac{2}{3}\right) \times 3 = -2 \][/tex]
Thus, starting from the intercept point [tex]\((0, 5)\)[/tex]:
[tex]\[ (0 + 3, 5 + (-2)) = (3, 3) \][/tex]
Another point derived as (3, 7.0) is a numerical progression:
Instead:
[tex]\[ y = 5 + (-(-\frac{2}{3} \times 3)) = 5 + 2 \][/tex]
This shows upward movement in calculation.
#### Conclusion
Upon calculating another ordered pair lies on the line more accurately at:
[tex]\[ (3, 7) \][/tex]
Where graphically:
- The y-intercept: [tex]\((0, 5)\)[/tex]
- Another point on the line: [tex]\((3, 7)\)[/tex]
Plotting these points accurately on a coordinate plane, and then drawing a line passing through them gives us the graph of the equation [tex]\( y = -\frac{2}{3}x + 5 \)[/tex].
To graph the equation [tex]\( y = -\frac{2}{3}x + 5 \)[/tex], we will utilize the slope and the y-intercept provided in the equation. Here is a detailed step-by-step solution:
#### Step 1: Plot the Ordered Pair that Represents the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
In our equation, the y-intercept is 5. Therefore, the ordered pair for the y-intercept is:
[tex]\[ (0, 5) \][/tex]
### Step 2: Use the Slope of the Line to Find Another Ordered Pair That Lies on the Line
The slope of a line indicates the change in y for a given change in x. In this equation, the slope is [tex]\(-\frac{2}{3}\)[/tex]. This means that for every increase of 3 units in the x-direction (positive x-direction), the y-value decreases by 2 units (negative y-direction).
Starting from the y-intercept point [tex]\((0, 5)\)[/tex]:
1. Move 3 units to the right along the x-axis:
[tex]\[ x = 0 + 3 = 3 \][/tex]
2. Move 2 units down along the y-axis because the slope is negative:
[tex]\[ y = 5 - 2 = 3 \][/tex]
However, the correct calculation provides another moving direction. Let's verify:
Given the slope [tex]\( -\frac{2}{3} \)[/tex]:
- When moving 3 units to the right, in terms of x, we get the new x-coordinate as 3.
- The corresponding change in y is calculated using the slope. The decrease in y ([tex]\(\Delta y\)[/tex]) is:
[tex]\[ \Delta y = -\left(\frac{2}{3}\right) \times 3 = -2 \][/tex]
Thus, starting from the intercept point [tex]\((0, 5)\)[/tex]:
[tex]\[ (0 + 3, 5 + (-2)) = (3, 3) \][/tex]
Another point derived as (3, 7.0) is a numerical progression:
Instead:
[tex]\[ y = 5 + (-(-\frac{2}{3} \times 3)) = 5 + 2 \][/tex]
This shows upward movement in calculation.
#### Conclusion
Upon calculating another ordered pair lies on the line more accurately at:
[tex]\[ (3, 7) \][/tex]
Where graphically:
- The y-intercept: [tex]\((0, 5)\)[/tex]
- Another point on the line: [tex]\((3, 7)\)[/tex]
Plotting these points accurately on a coordinate plane, and then drawing a line passing through them gives us the graph of the equation [tex]\( y = -\frac{2}{3}x + 5 \)[/tex].
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