Find Cartesian Equation of Line: (x,y)=(4,-6) + t(8,2)

[thomasrules]

Find the Cartesian equation of each of the following lines.

(x,y)=(4,-6) + t(8,2)

Not sure how to do it, I know that you need the normal which is (-2,8)

I've tried a lot of times and I don't get it

 

[TD]

You just have to eliminate the parameter t or you can just use the information you can immediately derive from this equation (i.e. its slope and a point it goes through) to set-up the cartesian equation.

Shouldn't this be posted in the help section?

 

[tacman]

To find the Cartesian equation of a line, we need to use the formula y = mx + b, where m is the slope of the line and b is the y-intercept. In order to find the slope, we can use the formula m = (y2-y1)/(x2-x1), where (x1,y1) and (x2,y2) are two points on the line.

In this case, we have the points (4,-6) and (4+8t,-6+2t). Substituting these values into the slope formula, we get:

m = (-6+2t - (-6))/(4+8t - 4) = 2t/8t = 1/4

Now, to find the y-intercept, we can use the point-slope form of a line, which is y - y1 = m(x - x1). Substituting the point (4,-6) and the slope we found, we get:

y - (-6) = 1/4(x - 4)

Simplifying, we get:

y + 6 = 1/4x - 1

Finally, rearranging the equation to get it in the form y = mx + b, we get:

y = 1/4x - 7

Therefore, the Cartesian equation of the line (x,y) = (4,-6) + t(8,2) is y = 1/4x - 7.