what is the equation in point-slope form of the line through the points (10,3) and (12,11)

Answer:Using the first point (10,3) the point-slope form gives: [tex]y-3=4(x-10)[/tex]Using the second point (12,11) the point-slope form gives: [tex]y-11=4(x-12)[/tex]Step-by-step explanation:Notice that you are given two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] on the plane through which the line has to go through.We can start then by finding the value of the slope for a segment that joins such points via the equation for the slope: [tex]slope=\frac{y_2-y_1}{x_2-x_1}[/tex]. Which in our case, if we call  [tex](x_1,y_1)[/tex] = (10,3) and [tex](x_2,y_2)[/tex]  = (12,11) give us:[tex]slope=\frac{y_2-y_1}{x_2-x_1}\\slope=\frac{11-3}{12-10}\\slope=\frac{8}{2}\\slope=4[/tex]Now that we have the slope of the line, we can write the "point-slope" form of it by using the information of on the general form of a line of slope "m" going through the point [tex](x_0,y_0)[/tex] in point-slope form:[tex]y-y_0=m(x-x_0)[/tex]we know our slope must be "4", and we can use any of the given points (for example (10,3) as the specific point [tex](x_0,y_0)[/tex], resulting in:[tex]y-y_0=m(x-x_0)\\y-3=4(x-10)[/tex]Of course, we could have used the other point as well, which would give us the following:[tex]y-y_0=m(x-x_0)\\y-11=4(x-12)[/tex]and although they look like different equations, they basically represent the very same equation, fact that we can verify by solving for "y" in both expressions:[tex]y-3=4(x-10)\\y-3=4x-40\\y=4x-40+3\\y=4x-37[/tex][tex]y-11=4(x-12)\\y-11=4x-48\\y=4x-48+11\\y=4x-37[/tex]