perpendicular or not

Question

without graphing tell whether the graphs of each pair of equations are perpendicular 2x-5y= -3

                                                                                                                          2x+5y=4.

Answer 1

Put each equation in the form y = mx+b.  Then slope = m

 

2x-5y = -3 → -5y = -2x-3 → y = (2/5)x + (3/5)    slope = 2/5

 

2x+5y = 4 → 5y = -2x+4 →y = (-2/5)x + (4/5)    slope = -2/5

 

Since the slopes of the two lines are not negative reciprocals, the lines are NOT perpendicular.  

Answer 2

If two lines are perpendicular, the product of their slopes must be -1.

 

2x - 5y= -3 ⇒ y = (2/5)x + 3/5 ⇒ m₁ = 2/5
2x + 5y = 4 ⇒ y = (-2/5)x + 4/5 ⇒ m₂ = -2/5

 

m₁m₂ = (2/5)(-2/5) = -4/25 ≠ -1

 

The lines are not perpendicular. 

Answer 3

Perpendicular lines have slopes that are the negative inverse of each other.  That is, if one line has slope m (the m in y=mx+b), then the slope of the perpendicular line is (-1/m).

 

When the equation of a line is given in Standard Format (Ax+By=C), the slope, m, equals (-A/B).

 

The problem gives two lines with slopes:

    -(2)/(-5)     and   -(2)/(5)

       2/5          and        -2/5

 

They are not perpendicular (note: not negative inverses).