Respuesta :
Question:
write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the equation.
(4,-3); y= 3x-5
Solution:
By definition, the slope-intercept form of a line is given by the following equation:
y = mx+b
where m is the slope of the line, and b is the y-coordinate of the y-intercept of the line. Now, if we have a perpendicular line to the line given by the equation y = mx+b, then we have that this perpendicular line has a slope 1/m (the reciprocal of the m).
Then, if we have a line given by equation
y = 3x-5
its slope is m = 3. And therefore, the slope of the line perpendicular to the above is m = 1/3. Then the slope-intercept of the perpendicular line is:
EQUATION 1:
[tex]y\text{ = }\frac{1}{3}x+b[/tex]now, we need to find b. To find it take a point that passes through the line, for example (x,y) = (4,-3), and replace it in the above equation:
[tex]-3\text{ = }\frac{1}{3}(4)+b[/tex]now, solve for b:
[tex]b\text{ = -3 - }\frac{4}{3}=\text{ }\frac{-13}{3}[/tex]Finally, replacing this value in equation 1, we obtain the slope-intercept form for the line:
[tex]y\text{ = }\frac{1}{3}x-\frac{13}{3}[/tex]Then, we can conclude that the correct answer is:
[tex]y\text{ = }\frac{1}{3}x-\frac{13}{3}[/tex]