Question

A line passes through `(4 ,1 )` and `(6 ,4 )`. A second line passes through `(3 ,5 )`. What is one other point that the second line may pass through if it is parallel to the first line?

Answer 1

`(1,2),(5,8)" and so on"`

Explanation:

The following result should be known.

`color(blue)"Parallel lines have equal slopes"`

To calculate the slope use the `color(blue)"gradient formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(m=(y_2-y_1)/(x_2-x_1))color(white)(2/2)|)))`
where m represents the slope and `(x_1,y_1),(x_2,y_2)" 2 points on the line"`

The 2 points here are (4 ,1) and (6 ,4)

let `(x_1,y_1)=(4,1)" and " (x_2,y_2)=(6,4)`

`rArrm=(4-1)/(6-4)=3/2`

Establish the equation of the line going through (3 ,5)

The equation of the line in `color(blue)"point-slope form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y-y_1=m(x-x_1))color(white)(2/2)|)))`
where m is the slope and `(x_1,y_1)" a point on the line"`

`"Using "m=3/2" and " (x_1,y_1)=(3,5)`

`y-5=3/2(x-3)larrcolor(red)"in point-slope form"`

distributing and simplifying gives an alternative version.

`y-5=3/2x-9/2`

`rArry=3/2x-9/2+5`

`rArry=3/2x+1/2larrcolor(red)" in slope-intercept form"`

Selecting values for x and substituting into the equation will give corresponding values of y, and hence coordinate points.

`•x=1toy=3/2+1/2=2rArr(1,2)" is a point on the line"`

`•x=5toy=15/2+1/2=8rArr(5,8)" is a point on the line"`
graph{3/2x+1/2 [-10, 10, -5, 5]}