Question

Line L has equation `2x- 3y=5`. Line M passes through the point (3, -10) and is parallel to line L. How do you determine the equation for line M?

Answer 1

See a solution process below:

Explanation:

Line L is in Standard Linear form. The standard form of a linear equation is: `color(red)(A)x + color(blue)(B)y = color(green)(C)`

Where, if at all possible, `color(red)(A)`, `color(blue)(B)`, and `color(green)(C)`are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

`color(red)(2)x - color(blue)(3)y = color(green)(5)`

The slope of an equation in standard form is: `m = -color(red)(A)/color(blue)(B)`

Substituting the values from the equation into the slope formula gives:

`m = color(red)(-2)/color(blue)(-3) = 2/3`

Because line M is parallel to line L, Line M will have the same slope.

We can now use the point-slope formula to write an equation for Line M. 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 point in the problem gives:

`(y - color(red)(-10)) = color(blue)(2/3)(x - color(red)(3))`

`(y + color(red)(10)) = color(blue)(2/3)(x - color(red)(3))`

If necessary for the answer we can transform this equation to the Standard Linear form as follows:

`y + color(red)(10) = (color(blue)(2/3) xx x) - (color(blue)(2/3) xx color(red)(3))`

`y + color(red)(10) = 2/3x - 2`

`color(blue)(-2/3x) + y + color(red)(10) - 10 = color(blue)(-2/3x) + 2/3x - 2 - 10`

`-2/3x + y + 0 = 0 - 12`

`-2/3x + y = -12`

`color(red)(-3)(-2/3x + y) = color(red)(-3) xx -12`

`(color(red)(-3) xx -2/3x) + (color(red)(-3) xx y) = 36`

`color(red)(2)x - color(blue)(3)y = color(green)(36)`