How do you find the x and y intercepts for #9x = 54 - 6y#?

To make the #y# term positive multiply everything on both sides by (-1) giving:

#-9x^2=-54+6y#

Isolate the #y# term: Add 54 to both sides

#54-9x^2=6y#

Isolate #y#: Divide throughout by 6

#54/6-9/6x=y#

Write as per convention

#y=-9/6x^2+54/6#

#y=-3/2x^2+9#
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#color(blue)("Determine the y-intercept")#

This occurs at #x=0#

#y_("intecpt")=-3/2(0^2)+9 = 9#
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#color(blue)("Determine the x-intercept")#

This occurs at #y=0#

#x_("intecpt")->y=0=-3/2x^2+9#

#3/2x^2=9#

#x^2=9xx2/3 = 6#

#x=-sqrt6 and x=+sqrt6#

Multiply both sides by (-1)

#-9x=-54+6y#

Add 54 to both sides

#-9x+54=6y#

Divide both side by 6

#-9/6x+54/6=y#

#y=-3/2x+9#
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#color(blue)("Determine the y-intercept")#

This occurs at #x=0#

#y=-3/2(0)+9=9#

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#color(blue)("Determine the x-intercept")#

This occurs at #y=0#

#y=0=-3/2x+9#

Add #3/2x# to both sides. Moves it to the left of the = sign.

#+3/2x=9#

Multiply both sides by #2/3#

#3/2xx2/3xx x=9xx2/3#

Using the principle that #2xx3=6=3xx2 -># you can move them around.

#3/3xx2/2xx x=9xx2/3#

#1color(white)("d")xx1xx x=9xx2/3#

#x_("intercept")=6#

color(white)("d")