Find the equations of line joining points #(-2,3)# and #(1,4)#?

3 Answers
Nov 15, 2016

The equation for line joining two points #(x_1,y_1)# and #(x_2,y_2)# is given by #(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)# and for given points it is #y=1/3x+11/3#

Explanation:

Let the slope intercept form of equation be #y=mx+c#

here we do not know the slope #m# and #y#-intercept #c#

What we know is that this passes through the two coordinate pairs, say #(x_1,y_1)# and #(x_2,y_2)#.

As such we have three equations

#y=mx+c# ......(1)

#y_1=mx_1+c# ......(2) and

#y_2=mx_2+c# ......(3)

Now using these let us eliminate #m# and #c#

subtracting (2) from (1), we get #(y-y_1)=m(x-x_1)# ......(4)

and subtracting (2) from (3), we get #(y-2-y_1)=m(x_2-x_1)# ......(5)

Dividing (4) by (5)

#(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)#

As the two points are #(-2,3)# and #(1,4)#, the equation is

#(y-3)/(4-3)=(x-(-2))/(1-(-2))#

or #(y-3)/1=(x+2)/(1+2)#

or #y-3=(x+2)/3#

or #3y-9=x+2#

or #3y=x+11# or #y=1/3x+11/3#

Nov 15, 2016

For any point on the line, the coordinate pair in slope-intercept form is#( x, x/3+11/3)#

Explanation:

For slope m and intercept c, the equation is y = m x +c.

The slope intercept form for coordinates is (x, m x +c ).

The slope of the line through the given points is

#m = ( 4-3 ) / (1-(-2))=1/3#

Also, from (1, 4). 4 = i/3(1) + c. So, c = 11/3.

So, the answer is #( x, 1/3x+11/3)#.

Nov 15, 2016

The equation of the line is:

#y= 1/3x +11/3# which can be written as #y = 1/3x +3 2/3#

Explanation:

If you are given the coordinates of 2 points on a line, substituting them into the formula below allows you to find the equation immediately. In the process you also calculate the slope.

#(-2,3) and (1,4)#
#(x_1,y_1) and (x_2,y_2)#

#color(red)((y-y_1)/(x-x_1) = (y_2-y_1)/(x_2-x_1))" "larr RHS = m#

#(y-3)/(x-(-2))= (4-3)/(1-(-2)) = 1/3#

#(y-3)/(x+2) = 1/3" "larr# now cross-multiply

#3(y-3) = x+2color(white)(xxxxxxxxxx)or y-3 = 1/3(x+2)#

#3y -9 = x+2color(white)(xxxxxxxxxxxxxxxxx) y = 1/3x+2/3+3#

#3y = x+11color(white)(xxxxxxxxxxxxxxxxxxx) y = 1/3x+3 2/3#

#y= 1/3x +11/3#