Find t such that (t,5) lies on the line through (0,3) and (-8,0)
slopes must be the same between any 2 points, so
(5-3)/(t-0) = (3-0)/(0+8)
2/t = 3/8
3t = 16
t = 16/3
Well, instead of finding t directly, let's have some fun and take a little detour! So, there were two points having an argument about who is closer to the line: (0,3) and (-8,0). The point (0,3) said, "I'm definitely closer!" while the point (-8,0) chimed in, "No way, I'm closer!"
Then, out of nowhere, the point (t,5) jumped in and said, "Hey, guys! Why don't you let me join in this crazy competition? I want to be closer to the line too!"
So, to fairly determine whether (t,5) is indeed closer, we'll have to find the slopes of the line between (0,3) and (-8,0) as well as between (0,3) and (t,5). If the slopes are equal, then both (0,3) and (t,5) must be equally close to the line!
Let's start by finding the slope between (0,3) and (-8,0). The slope (m) is given by the formula:
m = (y2 - y1) / (x2 - x1).
Plugging in the values, we get:
m = (0 - 3) / (-8 - 0) = -3 / -8 = 3/8.
Now, let's find the slope between (0,3) and (t,5). Again, using the same formula:
m = (5 - 3) / (t - 0) = 2 / t.
For (t,5) to lie on the same line, the slopes should be equal. So, we can set up an equation:
3/8 = 2 / t.
To solve this equation, we can cross multiply:
(3/8) * t = 2.
Now, let's bring out the clown calculator and do some number crunching:
3t /8 = 2.
3t = 2 * 8.
3t = 16.
t = 16 / 3.
So, the point (t,5) lies on the line through (0,3) and (-8,0) when t equals 16/3. Or in other words, (16/3, 5) is the lucky point that is equally close to the line as both (0,3) and (-8,0).
To find the value of t such that (t, 5) lies on the line through (0, 3) and (-8, 0), we need to determine the equation of the line first using the slope-intercept form, which is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Let's calculate the slope (m) of the line using the formula:
m = (y2 - y1) / (x2 - x1)
Using the coordinates (0, 3) and (-8, 0), we have:
m = (0 - 3) / (-8 - 0)
m = -3 / -8
m = 3/8
So, the slope (m) is 3/8.
Now, substitute one of the points (0, 3) into the equation y = mx + b to find the value of b (the y-intercept):
3 = (3/8)(0) + b
Simplifying this equation, we have:
3 = b
Therefore, the y-intercept (b) is 3.
Now that we know the slope (m) and the y-intercept (b), we can write the equation of the line as:
y = (3/8)x + 3
To find the value of t such that (t, 5) lies on the line, we substitute y = 5 into the equation and solve for x:
5 = (3/8)t + 3
Subtracting 3 from both sides of the equation, we have:
5 - 3 = (3/8)t
2 = (3/8)t
To isolate t, we multiply both sides of the equation by 8/3:
(8/3)(2) = t
16/3 = t
Therefore, t = 16/3.
So, the value of t such that (t, 5) lies on the line through (0, 3) and (-8, 0) is t = 16/3.
To find the value of t such that the point (t,5) lies on the line passing through (0,3) and (-8,0), we can use the slope-intercept form of a linear equation.
The slope-intercept form of a linear equation is given by: y = mx + b, where m represents the slope of the line and b represents the y-intercept.
First, let's find the slope (m) of the line using the given points (0,3) and (-8,0).
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values of the two points into the formula, we have:
m = (0 - 3) / (-8 - 0)
m = -3 / -8
m = 3/8
So, the slope (m) of the line is 3/8.
Now that we have the slope (m), we can use the point-slope form of the equation to find the equation of the line. The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) is any point on the line.
Using the point (0,3) and the slope (m = 3/8), we have:
y - 3 = (3/8)(x - 0)
y - 3 = (3/8)x
To find t such that the point (t,5) lies on this line, we substitute y = 5 into the equation and solve for t:
5 - 3 = (3/8)t
Now, solve for t:
2 = (3/8)t
To isolate t, we multiply both sides of the equation by 8/3:
t = (8/3)*2
Finally, simplify the expression:
t = 16/3
Therefore, the value of t such that the point (t,5) lies on the line passing through (0,3) and (-8,0) is t = 16/3.