Question

A line contains L(-4, -4) and M(2,3). Line q is in the same coordinate plane but does not intersect LM. Line q contains point N.

Answer 1

A general line equation can be written as:

$y = a*x + b$

Where a is the slope and b is the y-intercept.

Here we will find that for N(x₁, y₁), the line q is:

$y = \frac{7}{6}*x + (y_1 - \frac{7}{6}*x_1)$

If we know that the line passes through the points (x₁, y₁) and (x₂, y₂), then we can write the slope as:

$a = \frac{y_2 - y_1}{x_2 - x_1}$

In this case, we know that the line passes through (-4, -4) and (2, 3), then the slope is:

$a = \frac{3 - (-4)}{2 - (-4)} = \frac{7}{6}$

So the line is something like:

$y = \frac{7}{6} *x + b$

To find the value of b, we can use one of the two points, for example (2, 3) means that when x = 2, y must be equal to 3, then we can replace these two in the line equation to get:

$3 = \frac{7}{6}*2 + b\\\\3 = \frac{7}{3} + b\\\\3 - \frac{7}{3} = b = 2/3$

Then the line LM is;

$y = \frac{7}{6}*x + \frac{2}{3}$

Now we know that line q is in the same coordinate plane, but does not intersect LM. This means that these lines are parallel, and remember that two parallel lines have the same slope, then the slope of line q is also 7/6.

Now we know that line q passes through point N, but we do not know the coordinates of point N, then let's use general coordinates as N(x₁, y₁)

Then line q will be:

$y = \frac{7}{6}*x + c$

To find the value of c we use the point N.

$y_1 = \frac{7}{6}*x_1 + c\\\\y_1 - \frac{7}{6}*x_1 = c$

Then the equation of line q is:

$y = \frac{7}{6}*x + (y_1 - \frac{7}{6}*x_1)$

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