color(brown)("Extensive explanation given - using first principles method")
color(purple)("Note that the shortcut method is based on the outcome of first principle method")
Target: Manipulate the given equation into the form of y=mx+c
" "where m is the gradient and c is the y-intercept.
The 3/2y is positive so we need to keep the y on that side of the equation.
To get rid of the x on the left we change it to 0
What we do to one side of the equation we also do to the other.
Add color(red)(x) to both sides.
color(green)(3/2y-xcolor(red)(+x)" "=" "2color(red)(+x)
But -x+x=0
color(green)(3/2y+0=2+x)
Note that x+2 has the same value as 2+x
color(green)(3/2y=x+2
To 'get rid' of the 3/2 change it to 1 as 1xxy=y
Multiply both sides by color(red)(2/3)
color(green)(3/2color(red)(xx2/3)xxy=color(red)(2/3)(x+2)
color(green)(y=2/3x+4/3 rarr" "2/3" is the gradient (slope)"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The y-axis crosses the x-axis at x=0 so substitute 0 for x
y_("intercept")=2/3(0)+4/3" "=" "4/3
The x-axis cross the y-axis at y=0 so substitute 0 for y
0=2/3x+4/3
Subtract from both sides 4/3
-4/3=2/3x
Multiply both sides by 3/2
x=3/2xx(-4/3)
x=- (3/3xx4/2)
x_("intercept")=-2
