Verify if the lines y = 2x/3 + 8/3 and y = 3x + 5 are intercepting or they are parallel .
The give lines are y = 2x/3+8/3 (1) and
y = 3x+5.....(2).
Both lines are int he slope intercept form y = mx+c, where m is the slope.
Therefore the slopes of the lines are: 2/3 (1st line) and 3 (second line).
Since slopes are different , lines are not parallel and so they must intersect.
We now solve the equation by putting y = 3x+5 in the 1st equation:
3x+5 = 2x/3 +8/3.
Multiply by 3:
9x+15 = 2x+8
9x-2x = 8-15
7x = -7
x = -1.
Put x = 1 in the 2nd equation: y = 3(-1)+5 =2.
Therefore x = -1 and y = 2. Or (x,y ) = (-1 , 2) is the intersecting point of the lines.
Since the equations are put in the standard form:
y = mx + n
we can say that they are not parallel.
Let's see why?
According to the rule, 2 parallel lines have their slopes equal.
m1 = m2
We'll determine m1 and m2 and we'll notice that:
m1 = 2/3 and m2 = 3
They are not equal so the lines aren't parallel.
To verify if the lines have an intercepting point, we'll have to solve the system formd by the equations of the functions f and g.
The system will be:
y=2x/3 +8/3 (1)
We'll solve the system using the elimination method.
We'll subtract (2) from (1):
2x/3 + 8/3 - 3x - 5 = 0
2x + 8 - 9x - 15 = 0
9x - 2x = -15+8
We'll eliminate like terms:
7x = -7
We'll divide by -7:
x = -1
We'll substitute the value of x into (2):
y = -3+5
y = 2
The lines have an intercepting point and the coordinates of the intercepting point, (x,y), represent the solution of the system: (-1 , 2).