Which of the following best describes the graph of the system of equations shown below?
\[
6x - 14y = -28
\]
\[
3y - 7x = -14
\]
A. The lines are parallel.
B. The lines are the same.
C. The lines intersect but are not perpendicular.
D. The lines intersect and are perpendicular.
Answer (3)
k : 6 x − 14 y = − 28 ∣ s u b t r a c t 6 x f ro m b o t h s i d es − 14 y = − 6 x − 28 ∣ d i v i d e b o t h s i d es b y ( − 14 ) y = − 14 − 6 x − − 14 28 y = 7 3 x + 2 − − − − − − − − − − − − − − − − − − − − − − − − − − − − l : 3 y − 7 x = − 14 ∣ a dd 7 x t o b o t h s i d es 3 y = 7 x − 14 ∣ d i v i d e b o t h s i d es b y 3 y = 3 7 x − 3 14
Tw o l in es a re p er p e n d i c u l a r i f p ro d u c t o f t h e s l o p es i s e q u a l − 1. k : y = 7 3 x + 2 → t h e s k o lp e m k = 7 3 l : y = 3 7 x − 3 14 → t h e s l o p e m l = 3 7 m k × m l = 7 3 × 3 7 = 1 = − 1 co n c l u s i o n : t h e l in es a re n o t p er p e n d i c u l a r Tw o l in es a re p a r a ll e l i f t h e s l o p es a re e q u a l . m k = 7 3 ; m l = 3 7 → m k = m l co n c l u s i o n : t h e l in es a re n o t p a r a ll e l
A n s w er : C − T h e l in es in t ersec t b u t a re n o t p er p e n d i c u l a r .
First, you want to find out what the slope and the y-intercept is, you must put both equations in y=mx+b from.
6x − 14y = −28 :First, subtract the 6x over to the right -6x -6x
( -14y = -6x - 28 ) ÷ -14 :Then, divide the whole equation by -14
y = 6/14x + 2 :Now, reduce the fraction, divide both the top and the bottom by 2
y = 3/7x + 2 3y − 7x = −14 :First, add the 7x over to the right side +7x +7x
(3y = 7x - 14) ÷ 3 :Next, divide the whole equation by 3
***y = 7/3x - 14/3
y = 3/7x + 2 slope: 3/7 y-int.: 2
y = 7/3x - 14/3 slope: 7/3 y-int.: -14/3 or -4 7/10 or -4.7
C. Your answer is C. They will eventually intersect, but not at a right angle.
The lines represented by the equations intersect but are not parallel or perpendicular. The slopes of the two lines are different, confirming that they meet at one point. Hence, the best description is that the lines intersect but are not perpendicular.
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