A line can be expressed in the standard form y = mx + c where m is the slope and c is the y-intercept. For the two line to be parallel they should have the same slope.
Now, y = 7x / 2 - 17/2 has the slope m equal to 7/2
For y = 14x/9 - 24/9 , the slope m is equal to 14/9
Now we see that the slopes of the lines which are 14/9 and 7/2 are not equal. Therefore the two lines are not parallel.
Two lines that are not parallel are intersecting but you do not want the point of intersection, instead you just want to know if they are parallel or not.
Therefore the lines are not parallel.
To prove that the lines d1 and d2 are parallel we'll have to verify if the system formed from the equations of d1 and d2 has not any solutions.
We'll put the equation of the lines in the general form:
ax + by + c = 0
The first equation is:
The second equation is:
We'll form the system:
We'll add 17 both sides:
7x - 2y = 17 (1)
We'll add 24 both sides:
14x - 9y = 24 (2)
We'll solve the system using elimination method. For this reason, we'll multiply (2) by -2 and we'll add the resulting equation to (1):
-14x + 4y = -34 (3)
(1) + (3): 14x - 9y - 14x + 4y = 24 - 34
We'll eliminate and combine like terms:
-5y = -10
We'll divide by -5:
y = 2
We'll substitute y in (1):
14x - 9y = 24
14x - 18 = 24
14x = 24 + 18
14x = 42
7x = 21
x = 3
The system has a solution that represents the intercepting point of the lines: (3,2). The given lines are not parallel but they are intercepting.
To show that lines y = 7x/2 - 17/2 and y = 14x/9 - 24/9.
We know that y = mx+c is the slope intercept form of the equation of a line. Here m is the slope.
If the slopes are two lines are equal , then they are paralle.
The line y = 7x/2-17/2 is in the slope intercept form So it has the slope m = 7/2.
Similarly the line y = 14x/9-24/9 is in the slope intercept form. So the line has the slope 14/9
So the slopes of the two lines are 7/2 = 3.5 and 14/9 = 1.6667. So both lines have the different slopes. So the lines y = 14x/2 -17/2 and y = 14x/9 -24/9 are not parallel.