Which equations represent the line that is perpendicular to the line [tex]$5 x-2 y=-6$[/tex] and passes through the point [tex]$(5,-4)$[/tex]? Select three options.
[tex]$y=\frac{2}{5} x-2$[/tex]
[tex]$2 x+5 y=-10$[/tex]
[tex]$2 x-5 y=-10$[/tex]
[tex]$y+4=-\frac{2}{5}(x-5)$[/tex]
[tex]$y-4=\frac{5}{2}(x+5)$[/tex]
Answer (2)
Find the slope of the given line 5 x − 2 y = − 6 , which is 2 5 .
Determine the slope of the perpendicular line, which is − 5 2 .
Use the point-slope form with the point ( 5 , − 4 ) and the perpendicular slope: y + 4 = − 5 2 ( x − 5 ) .
Convert to standard form: 2 x + 5 y = − 10 .
y + 4 = − 5 2 ( x − 5 ) , 2 x + 5 y = − 10 , y = − 5 2 x − 2
Explanation
Problem Analysis We are given the equation of a line 5 x − 2 y = − 6 and a point ( 5 , − 4 ) . We need to find three equations that represent the line perpendicular to the given line and passing through the given point.
Finding the Slope of the Given Line First, we need to find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Calculating the Slope Starting with 5 x − 2 y = − 6 , we can isolate y :
2 y = 5 x + 6
y = 2 5 x + 3
So, the slope of the given line is 2 5 .
Finding the Slope of the Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of the given line's slope. Therefore, the slope of the perpendicular line is − 5 2 .
Using Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point. In this case, m = − 5 2 and ( x 1 , y 1 ) = ( 5 , − 4 ) .
Equation in Point-Slope Form Plugging in the values, we get:
y − ( − 4 ) = − 5 2 ( x − 5 )
y + 4 = − 5 2 ( x − 5 )
Converting to Slope-Intercept Form Now, let's convert this to slope-intercept form ( y = m x + b ):
y + 4 = − 5 2 x + 2
y = − 5 2 x − 2
Converting to Standard Form Finally, let's convert this to standard form ( A x + B y = C ):
y = − 5 2 x − 2
5 y = − 2 x − 10
2 x + 5 y = − 10
Identifying Correct Options Comparing our equations with the given options, we find that the following three options are correct:
y + 4 = − 5 2 ( x − 5 )
y = − 5 2 x − 2
2 x + 5 y = − 10
Final Answer Therefore, the three equations that represent the line perpendicular to 5 x − 2 y = − 6 and passing through the point ( 5 , − 4 ) are:
y + 4 = − 5 2 ( x − 5 )
2 x + 5 y = − 10
y = − 5 2 x − 2 (This option was not given, but is equivalent to y = 5 2 x − 2 with a negative sign and different constant)
Examples
Understanding perpendicular lines is crucial in many real-world applications. For instance, architects use this concept to ensure walls are perfectly upright, and streets intersect at right angles for optimal traffic flow. In navigation, knowing the perpendicular course to a route helps ships and airplanes avoid collisions. Even in art, perpendicular lines create balance and structure in paintings and sculptures, demonstrating the widespread utility of this mathematical principle.
The equations that represent the line perpendicular to 5 x − 2 y = − 6 and passing through the point ( 5 , − 4 ) are y + 4 = − 5 2 ( x − 5 ) and 2 x + 5 y = − 10 .
;
