Find the equation of the line through (2,a) and (5,b).

Question

Find the equation of the line through (2,a) and (5,b).

Y=

Answer 1

We need to find the slope and the y-intercept in terms of a and b; once we do, this equation will work for any combination of a and b to produce a line.

Slope:

Slope can be found by simply subtracting the two given points m=(y2-y1)/(x2-x1) ; m=(b-a)/3

y-intercept:

We will denote the intercept as "c" because the variable "b" is already taken, as it is usually called "b" in slope-intercept form. We can use the evaluated slope-intercept equations to solve for c. First we write the equations in slope-intercept form:

a=m2+c and b=m5+c

Next we solve for c in each equation: c=a-2m and c=b-5m

We then choose one equation to move forward with, let's say c=b-5m ; and we solved for m earlier, so:

c=b-5((b-a)/3) which you should simplify yourself to find:

c=(1/3)(5a-2b) (you should also verify this by choosing c=a-2m )

Putting it all together:

Y=((b-a)/3)x + (1/3)(5a-2b)

Answer 2

Use the two-point form:

(y - a) / (x - 2) = (a - b) / (2 - 5)