A line passes through #(5 ,8 )# and #(2 ,9 )#. A second line passes through #(3 ,5 )#. 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    ,8   )# and #(2    ,9   )#.  A second line passes through #(3    ,5  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Jayden Jackson · Accepted Answer

There are many answers based on the selected #x# value.
One answer is #(6, 4)# Given: two parallel lines: #"line"_1 (5, 8), (2, 9); " " "line"_2 (3,5)# Find the slope of #"line"_1#: #m = (y_2 - y_1)/(x_2 - x_1) = (9 - 8)/(2 - 5) = 1/-3 = -1/3# Parallel lines have the same slope. Find the equation of the second line using the given point: #y - y_1 = m(x - x_1)# #y - 5 = -1/3 (x - 3)# Simplify by using distribution and adding #5# to both sides: #y - 5 = -1/3x + (-1/3)(-3/1)# #y - 5 = -1/3x + 1# #y = -1/3x + 6# To get another point, just plug in any #x# value. Let #x = 6; y = -1/3 * 6/1 + 6 = -2 + 6 = 4; " "(6, 4)#

Caroline Aucoin · Answer

undefined To find a point that the second line may pass through if it is parallel to the first line, we need to use the slope of the first line. The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}$.

Using the given points (5, 8) and (2, 9) for the first line, the slope is $\frac{9 - 8}{2 - 5} = -\frac{1}{3}$.

Since the second line is parallel to the first line, it will have the same slope of $-\frac{1}{3}$. Using the point (3, 5) on the second line and the slope, we can find the y-intercept $b$ using the point-slope form of a line equation: $y = mx + b$, where $m$ is the slope.

Substituting the slope $-\frac{1}{3}$ and point (3, 5) into the equation gives:
$$5 = -\frac{1}{3} \times 3 + b$$
$$5 = -1 + b$$
$$b = 6$$

So, the equation of the second line is $y = -\frac{1}{3}x + 6$.

Now, to find another point on this line, we can choose any x-value and solve for y. For example, let's choose $x = 0$:
$$y = -\frac{1}{3} \times 0 + 6$$
$$y = 6$$

Therefore, another point that the second line may pass through is (0, 6).