Find the scalar equation of the line with vector equation (x,y)= (-1,2) + t(3,7), ter.
My friend and I got different answers, I got 0= 7x - 3y + 13, while she got 0= -7x+2y-11
two points on the line (for t=0,1) are (-1,2) and (2,9)
The slope of that line is thus 7/3, so the line will be
7x-3y+k = 0
or
-7x+3y + k = 0
for some k.
Both equations work for (-1,2) but not for (2,9)
picking up from oobleck ...
7x - 3y + k = 0
but (-1,2) lies on it, so
-7 - 6 + k = 0
k = 13
your equation is 7x - 3y + 13 = 0
testing the other point (2,9)
LS = 14 - 27 + 13 = 0 = RS
so both points work
To find the scalar equation of the line, we need to eliminate the parameter, t, from the vector equation. Let's do this step by step.
The vector equation (x, y) = (-1, 2) + t(3, 7) can also be written as:
x = -1 + 3t
y = 2 + 7t
To eliminate the parameter t, we can rewrite the equations to isolate t.
From the first equation: t = (x + 1) / 3
From the second equation: t = (y - 2) / 7
Now, set the two expressions for t equal to each other and solve for x and y:
(x + 1) / 3 = (y - 2) / 7
To remove the fractions, we can cross-multiply:
7(x + 1) = 3(y - 2)
Expanding the equation:
7x + 7 = 3y - 6
Rearranging the terms:
7x - 3y = -13
This is the scalar equation of the line. So, the correct answer is:
0 = 7x - 3y + 13
To find the scalar equation of the line with a given vector equation, we need to eliminate the parameter 't' from the equation.
The given vector equation is (x,y) = (-1,2) + t(3,7).
To eliminate 't', we can equate the x and y components of the equation separately.
The x-component equation is:
x = -1 + 3t
The y-component equation is:
y = 2 + 7t
To obtain the scalar equation, we need to express 't' in terms of x and y.
From the x-component equation, we can solve for 't':
t = (x + 1) / 3
Substituting this value of 't' into the y-component equation, we get:
y = 2 + 7[(x + 1) / 3]
Simplifying this equation further:
3y = 6 + 7(x + 1)
3y = 6 + 7x + 7
3y = 7x + 13
Dividing both sides by 3:
(7x - 3y) / 3 = -13 / 3
Hence, the scalar equation of the line is:
7x - 3y = -13
Therefore, your friend's equation 0 = -7x + 2y - 11 is not correct, while your equation 0 = 7x - 3y + 13 is correct.