Question

A line passes through `(3 ,2 )` and `(7 ,3 )`. A second line passes through `(8 , 1 )`. What is one other point that the second line may pass through if it is parallel to the first line?

Answer 1

Many possible answers, including `(12, 2)` and `(4, 0)`

Explanation:

Lines are parallel if they have the same slope.

So, to answer this question, we need to:
1. Find the slope of the first line
2. Apply that slope to the second line to find a corresponding point

SOLUTION

1. Slope of first line
   Slope or `m` can be found by: `m=(Delta y)/(Delta x)=(y_2 - y_1)/(x_2 - x_1)`
   Using the given information, `m=(3 - 2)/(7 - 3)=color(blue)(1/4)`
   Both lines will have a slope of `1/4`
2. Using `m=1/4`, find a point on the second line

Here, there are infinite possibilities for points on the second line.
If we use the point `(8,1)` as `(x_1, y_1)`, then `(x_2, y_2)` must
satisfy the equation

`m=1/4=(y_2 - y_1)/(x_2 - x_1)`

The easiest solutions can be found by using `Delta y=1` and `Delta x=4`

`Delta y = 1`
`y_2 - y_1 = 1`
`y_2 - 1 = 1`
`color(blue)(y_2 = 2)`

`Delta x = 4`
`x_2 - x_1 = 4`
`x_2 - 8 = 4`
`color(blue)(x_2 = 12)`

One solution is `(12, 2)`

Another solution can be found by solving
`1 - y = 1` and `8 - x = 4`

But as stated earlier, there are many, many more valid solutions.