Question

Find the equation of the line that passes through the midpoint of `(-5,6) and (9,8)` and is parallel to the line `5x-6y+4=0`?

Answer 1

`y=5/6x+16/3`

Explanation:

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Let's call the midpoint point `P`. We use the midpoint formula to find the coordinates of the midpoint of the first line:

`P(x,y)=P((x_1+x_2)/2,(y_1+y_2)/2)=P((-5+9)/2,(6+8)/2)`

`P(x,y)=P(2,7)`

Now, we have to write the equation of the line that goes through point `P(2,7)` and is parallel to the second line. For the line to be parallel to another line, it would have to have the same slope. Let's write the equation of the second line as:

`5x-6y+4=0`

`6y=5x+4`

`y=5/6x+2/3`

This is now in the form of:

`y=mx+b` where `m` is the slope and `b` is the `y`-intercept.

Therefore, the slope of the second line is `5/6` which will also be the slope of the line through `P`.

The equation of this line is:

`y=5/6x+b`

We can use the coordinates of the point `P` in this equation to solve for `b`:

`7=5/6(2)+b=5/3+b`

`b=7-5/3=(21-5)/3=16/3`

The equation of the line is:

`y=5/6x+16/3`