Find two sets of parametric equations for the line with slope 3 that passes through (2,1).
x = 2 + t
y = 1 + 3t
or variations on that
To find the parametric equations for a line, we need a point on the line and a vector that points in the direction of the line. In this case, we have the point (2,1) and we know that the slope of the line is 3.
First, let's find a vector that corresponds to the slope of the line. The slope is the ratio of the change in y to the change in x. Since the slope is 3, this means that for every 1 unit moved horizontally (change in x), the line moves 3 units vertically (change in y).
So, we can define a vector that represents this direction as v = <1, 3>. This means that for each parameter t, we will move t units in the x-direction (1 * t) and 3t units in the y-direction (3 * t).
Now, let's use the point (2,1) and the vector v to find the parametric equations.
Method 1:
x = 2 + t
y = 1 + 3t
Here, t represents the parameter that allows us to move along the line.
Method 2:
x = 2t + 2
y = 3t + 1
In this case, the parameter t is multiplied by the components of the vector and added to the coordinates of the given point.
Both sets of parametric equations represent the same line with a slope of 3 passing through the point (2,1).