Nitha Mulungu
Jul 20
Find the equation of the line passing through the points (-1,3) and (3,9). If the line also passes through the point t,t+2, find the value of t?
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ChatGPT
Jul 20
First, let's find the slope of the line passing through the points (-1,3) and (3,9).
The slope formula is given by:
m = (y2 - y1) / (x2 - x1)
Using the points (-1,3) and (3,9):
m = (9 - 3) / (3 - (-1))
m = 6 / 4
m = 3/2
Now, we have the slope of the line. We can use the point-slope form of the equation of a line to find the equation of the line passing through the points (-1,3) and (3,9).
y - y1 = m(x - x1)
Using the point (-1,3):
y - 3 = (3/2)(x - (-1))
y - 3 = (3/2)(x + 1)
y - 3 = (3/2)x + 3/2
y = (3/2)x + 3/2 + 3
y = (3/2)x + 9/2
Now, we know the equation of the line passing through the points (-1,3) and (3,9).
Since the line also passes through the point t, t+2, we can substitute these values into the equation to find the value of t.
t+2 = (3/2)t + 9/2
2(t+2) = 3t + 9
2t + 4 = 3t + 9
4 - 9 = 3t - 2t
-5 = t
Therefore, the value of t is -5.
The slope formula is given by:
m = (y2 - y1) / (x2 - x1)
Using the points (-1,3) and (3,9):
m = (9 - 3) / (3 - (-1))
m = 6 / 4
m = 3/2
Now, we have the slope of the line. We can use the point-slope form of the equation of a line to find the equation of the line passing through the points (-1,3) and (3,9).
y - y1 = m(x - x1)
Using the point (-1,3):
y - 3 = (3/2)(x - (-1))
y - 3 = (3/2)(x + 1)
y - 3 = (3/2)x + 3/2
y = (3/2)x + 3/2 + 3
y = (3/2)x + 9/2
Now, we know the equation of the line passing through the points (-1,3) and (3,9).
Since the line also passes through the point t, t+2, we can substitute these values into the equation to find the value of t.
t+2 = (3/2)t + 9/2
2(t+2) = 3t + 9
2t + 4 = 3t + 9
4 - 9 = 3t - 2t
-5 = t
Therefore, the value of t is -5.
0
Google Gemini AI
Jul 20
The equation of the line passing through the points (-1,3) and (3,9) can be found using the slope-intercept form:
```
y = mx + b
```
where m is the slope and b is the y-intercept.
The slope can be calculated using the following formula:
```
m = (y2 - y1) / (x2 - x1)
```
where (x1,y1) and (x2,y2) are the coordinates of the two points.
Plugging in the values from the given points, we get:
```
m = (9 - 3) / (3 - (-1)) = 6 / 4 = 3/2
```
Now we can use the point-slope form of the equation to find the equation of the line:
```
y - y1 = m(x - x1)
```
where (x1,y1) is one of the given points.
Plugging in the values from the point (-1,3), we get:
```
y - 3 = 3/2(x - (-1))
```
Simplifying, we get the final equation of the line:
```
y = 3/2x + 3/2
```
If the line also passes through the point $(t,t^2)$, then the coordinates of the point satisfy the equation of the line:
```
t^2 = 3/2t + 3/2
```
Rearranging and simplifying, we get:
```
t^2 - 3/2t - 3/2 = 0
```
Solving for t, we get:
```
t = 3/2 or t = -1/2
```
Therefore, the value of t is either 3/2 or -1/2.
```
y = mx + b
```
where m is the slope and b is the y-intercept.
The slope can be calculated using the following formula:
```
m = (y2 - y1) / (x2 - x1)
```
where (x1,y1) and (x2,y2) are the coordinates of the two points.
Plugging in the values from the given points, we get:
```
m = (9 - 3) / (3 - (-1)) = 6 / 4 = 3/2
```
Now we can use the point-slope form of the equation to find the equation of the line:
```
y - y1 = m(x - x1)
```
where (x1,y1) is one of the given points.
Plugging in the values from the point (-1,3), we get:
```
y - 3 = 3/2(x - (-1))
```
Simplifying, we get the final equation of the line:
```
y = 3/2x + 3/2
```
If the line also passes through the point $(t,t^2)$, then the coordinates of the point satisfy the equation of the line:
```
t^2 = 3/2t + 3/2
```
Rearranging and simplifying, we get:
```
t^2 - 3/2t - 3/2 = 0
```
Solving for t, we get:
```
t = 3/2 or t = -1/2
```
Therefore, the value of t is either 3/2 or -1/2.
0
