Equations of the line passing through 1,2,3 and 3 4 3 are

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First, we must know what an equation of a line looks like. We write the equation in slope-intercept form:

#y=mx+b#
(The#m#is the slope, and#b#is the y-intercept)

Next, find the slope (#m#) of the line by using the formula#(y_2-y_1)/ (x_2-x_1)#:

#((6)-(3))/((4)-(1))##=##3/3##=##1#

Next, find the y-intercept (#b#) by using the slope-intercept form equation and substituting#1#in for#m#and one of the ordered pairs in for#x#and#y#:

First, we need to determine the slope of the line. The slope can be found by using the formula:#m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where#m#is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.

Substituting the values from the points in the problem gives:

#m = (color(red)(-4) - color(blue)(2))/(color(red)(3) - color(blue)(-1)) = (color(red)(-4) - color(blue)(2))/(color(red)(3) + color(blue)(1)) = -6/4 = -3/2#

Next we can use the point-slope formula to write an equation for the line. The point-slope formula states:#(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#

Where#color(blue)(m)#is the slope and#(color(red)(x_1, y_1))#is a point the line passes through. Substituting the slope we calculated and the values from the second point in the problem gives:

#(y - color(red)(-4)) = color(blue)(-3/2)(x - color(red)(3))#

#(y + color(red)(4)) = color(blue)(-3/2)(x - color(red)(3))#

We can also substitute the slope we calculated and the values from the first point in the problem giving:

#(y - color(red)(2)) = color(blue)(-3/2)(x - color(red)(-1))#

#(y - color(red)(2)) = color(blue)(-3/2)(x + color(red)(1))#

We can solve this equation for#y#to write an equation in slope intercept form. The slope-intercept form of a linear equation is:#y = color(red)(m)x + color(blue)(b)#

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