A line passes through #(5 ,0 )# and #(7 ,3 )#. A second line passes through #(3 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?

Question

A line passes through #(5    ,0  )# and #(7   ,3   )#.  A second line passes through #(3    ,1  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Theodore Abad · Accepted Answer

see below

First, find the gradient of lines that across #(5,0)# and #(7,3)# using formula #m=(y_2-y_1)/(x_2-x_1)# #m=3/2# then gradient of two lines should be same since parallel. Then using straight line equation where is #y-y_1=m(x-x_1)# then we will get #y=3/2x-7/2# Use any value to assign into straight line equation. In this case, I will assume #x=0# then #y=-7/2# Conclusion the point is #(0,-7/2)#

Caroline Aucoin · Answer

undefined If the second line is parallel to the first line passing through points (5, 0) and (7, 3), then it must have the same slope as the first line.

The slope of the first line can be calculated using the formula:

$$ \text{Slope} = \frac{{\text{change in } y}}{{\text{change in } x}} = \frac{{3 - 0}}{{7 - 5}} = \frac{3}{2} $$

Since the second line is parallel to the first line, it must also have a slope of $ \frac{3}{2} $.

Now, using the point-slope form of a line equation, we can find the equation of the second line using the given point (3, 1):

$$ y - y_1 = m(x - x_1) $$

Substituting the values of $ m $, $ x_1 $, and $ y_1 $ into the equation:

$$ y - 1 = \frac{3}{2}(x - 3) $$

Now, solve this equation for $ y $ to find the equation of the second line.

$$ y = \frac{3}{2}x - \frac{9}{2} + 1 $$
$$ y = \frac{3}{2}x - \frac{9}{2} + \frac{2}{2} $$
$$ y = \frac{3}{2}x - \frac{7}{2} $$

So, the equation of the second line parallel to the first one is $ y = \frac{3}{2}x - \frac{7}{2} $.

Now, you can find another point on this line by choosing any value for $ x $ and then calculating the corresponding $ y $ value. For instance, if $ x = 1 $, then:

$$ y = \frac{3}{2}(1) - \frac{7}{2} = \frac{3}{2} - \frac{7}{2} = -2 $$

Thus, when $ x = 1 $, $ y = -2 $, giving us the point (1, -2) as another point through which the second line passes.