A line passes through #(3 ,9 )# and #(5 ,8 )#. A second line passes through #(7 ,6 )#. What is one other point that the second line may pass through if it is parallel to the first line?
1 Answer
(0, 19/2)
Explanation:
First, calculate the equation of the first line. To be parallel, the second line must have the same slope. Using the given third point the general equation can be written. Then any other set of points that satisfies that equation is a solution.
y = mx + b Calculate the slope, m, from the given point values, solve for b by using one of the point values, and check your solution using the other point values.
A line can be thought of as the ratio of the change between horizontal (x) and vertical (y) positions. Thus, for any two points defined by Cartesian (planar) coordinates such as those given in this problem, you simply set up the two changes (differences) and then make the ratio to obtain the slope, m.
Vertical difference “y” = y2 – y1 = 8 – 9 = -1
Horizontal difference “x” = x2 – x1 = 5 - 3 = 2
Ratio = “rise over run”, or vertical over horizontal = -1/2 for the slope, m.
A line has the general form of y = mx + b, or vertical position is the product of the slope and horizontal position, x, plus the point where the line crosses (intercepts) the x-axis (the line where x is always zero.) So, once you have calculated the slope you can put any of the two points known into the equation, leaving us with only the intercept 'b' unknown.
9 = (-1/2)(3) + b ; 9 = -(3/2) + b ; 9 + 3/2 = b ; 21/2 = b
Thus the final equation is y = -(1/2)x + 21/2
We then check this by substituting the other known point into the equation:
8 = (-1/2)(5) + 21/2 ; 8 = -5/2 + 21/2 ; 8 = 16/2 ; 8 = 8 CORRECT!
The second line must then have the form of y = -(1/2)x + b also.
Again, substituting in the given point we find the general equation.
6 = (-1/2)(7) + b ; 6 = -7/2 + b ; b = 19/2 Our equation is thus:
y = (-1/2)x + 19/2
Pick any (x,y) pair of points that solve this equation for the desired answer. ‘0’ works well for ‘x’…
y = (-1/2)0 + 19/2 ; y = 19/2
Therefore, our other point on the parallel line is (0, 19/2).
