Question

If the graphs of the lines 2x-5y=7 and 10x +by=7 are perpendicular, what is the value of b?

Answer 1

b = 4

Explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

2x - 5y = 7 ( subtract 2x from both sides )

- 5y = - 2x + 7 ( divide through by - 5 )

y =
`(2)/(5)` x -
`(7)/(5)` ← in slope- intercept form

with slope m =
`(2)/(5)`

given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/((2)/(5) )` = -
`(5)/(2)`

given

10x + by = 7 is perpendicular to the line 2x - 5y = 7

then line 10x + by = 7 has a slope m of -
`(5)/(2)`

subtract 10x from both sides

by = - 10x + 7 ( divide through by b , b ≠ 0 )

y = -
`(10)/(b)` x +
`(7)/(b)` ← in slope- intercept form

with slope m = -
`(10)/(b)`

equate -
`(10)/(b)` and -
`(5)/(2)` and solve for b

-
`(10)/(b)` = -
`(5)/(2)` ( cross- multiply )

5b =- 2 × - 10 = 20 ( divide both sides by 5 )

b = 4