A line passes through (4 ,3 ) and (7 ,1 ). A second line passes through (1 ,1 ). What is one other point that the second line may pass through if it is parallel to the first line?

May 1, 2018

The two lines have the same slope because they are parallel.

One point on the second line is $\left(\frac{3}{2} , 2\right)$.

Explanation:

In order to solve this question, we need to find the slope of the first line. The second will have the same slope, since they are parallel.

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
$= \frac{1 - 3}{7 - 4} = - \frac{2}{3}$

If we write the equation of the second line as:

$y = m x + c$ we can substitute in the value of $m$:

$y = \frac{2}{3} x + c$

We can substitute in the point $\left(1 , 1\right)$ to find the value of $c$:

$1 = \frac{2}{3} \left(1\right) + c$

$c = \frac{1}{3}$

So $y = \frac{2}{3} x + \frac{1}{3}$

Any point that we can substitute into this equation that makes it true is a point on that line. Let's choose $y = 2$ and find the value of $x$ that makes the equation too:

$2 = \frac{2}{3} x + \frac{1}{3}$

$x = \frac{3}{2}$

One point on the second line, then, is $\left(\frac{3}{2} , 2\right)$.