color(blue)(2y+3x-4=0)" the equation of the blue line"2y+3x−4=0 the equation of the blue line
"let's rearrange the equation of the blue line"let's rearrange the equation of the blue line
color(blue)(2y=-3x+4)2y=−3x+4
color(blue)(y=-3/2 x+2) " (1)"y=−32x+2 (1)
y=m x+n " represents the standard form of the line"y=mx+n represents the standard form of the line
"where m is slope of line"where m is slope of line
m=-3/2 m=−32
"let the slope of the red line perpendicular to the blue line be " m_1let the slope of the red line perpendicular to the blue line be m1
m*m1=-1m⋅m1=−1
-3/2*m_1=-1−32⋅m1=−1
m_1=2/3m1=23
"now we can use the slope-intercept form of the equation of line"now we can use the slope-intercept form of the equation of line
color(red)(y-y_1=m(x-x_1)y−y1=m(x−x1)
color(red)(y-3=2/3(x+2))y−3=23(x+2)
color(red)(y-3=2/3 x+4/3)y−3=23x+43
color(red)(y=2/3 x+4/3+3)y=23x+43+3
color(red)(y=2/3 x+13/3)y=23x+133
"or "or
"let's multiply both sides of equation by 3"let's multiply both sides of equation by 3
3y=2x+133y=2x+13
-2x+3y=13−2x+3y=13