A line passes through #(4 ,3 )# and #(1 ,4 )#. A second line passes through #(7 ,8 )#. 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 #(4    ,3  )# and #(1   ,4   )#.  A second line passes through #(7    ,8  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Noah Atkinson · Accepted Answer

Let the coordinate of other point be #(h,k)# The slope of the 1st line passing through #(4,3) and (1,4)# is #=(3-4)/(4-1)=-1/3# The slope of the 2nd line passing through #(h,k) and (7,8)# is #=(k-8)/(h-7)# By the given condition of the problem the 2nd line is  parallel
to the !st one . So their slope will be equal. #=>(k-8)/(h-7)=-1/3# #=>h-7=3k-24# #=>h-3k+17=0# Replacing #h->x and k->y# we get an equation of a straight line  #x-3y+17=0   ----(1)# 
#color(blue)(-> "Any point on this straight line may be our other point")#. Plugging in 20 for x in equation (1) we get y = 1, so #(color(red)(1,20))# may be the coordinate of other point among infinite number  of possibilities.

Ella Armstrong · Answer

undefined If the second line is parallel to the first line, it means that the slopes of the two lines are equal. The slope of the first line can be calculated using the formula:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

For the first line:
$$(4, 3)$$ and $$(1, 4)$$

$$m = \frac{{4 - 3}}{{1 - 4}} = \frac{{1}}{{-3}} = -\frac{{1}}{{3}}$$

Since the second line is parallel to the first, it must also have a slope of $-\frac{{1}}{{3}}$. Using the point-slope form of a line, we can find another point on the second line. Let's use the point $(7, 8)$ which is given.

The point-slope form of a line is:

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

Substitute $x_1 = 7$, $y_1 = 8$, and $m = -\frac{{1}}{{3}}$:

$$y - 8 = -\frac{{1}}{{3}}(x - 7)$$

$$y - 8 = -\frac{{1}}{{3}}x + \frac{{7}}{{3}}$$

$$y = -\frac{{1}}{{3}}x + \frac{{7}}{{3}} + 8$$

$$y = -\frac{{1}}{{3}}x + \frac{{7}}{{3}} + \frac{{24}}{{3}}$$

$$y = -\frac{{1}}{{3}}x + \frac{{31}}{{3}}$$

So, one other point that the second line may pass through if it is parallel to the first line is $(10, 21)$.