Given the equation:
[tex]\[
\frac{2}{3} x + c y = 2
\][/tex]
we need to determine the value of [tex]\( c \)[/tex] when the slope of this equation is 6.
First, let's rewrite the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]. To do this, we need to solve for [tex]\( y \)[/tex]:
[tex]\[
\frac{2}{3} x + c y = 2
\][/tex]
Subtract [tex]\(\frac{2}{3} x\)[/tex] from both sides to isolate the term involving [tex]\( y \)[/tex]:
[tex]\[
c y = -\frac{2}{3} x + 2
\][/tex]
Next, divide every term by [tex]\( c \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{2}{3c} x + \frac{2}{c}
\][/tex]
The slope-intercept form of the equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. From the equation above, we can see that the coefficient of [tex]\( x \)[/tex] is [tex]\( -\frac{2}{3c} \)[/tex]. Given that the slope [tex]\( m \)[/tex] is 6, we set up the equation:
[tex]\[
-\frac{2}{3c} = 6
\][/tex]
Now, solve for [tex]\( c \)[/tex]:
First, multiply both sides of the equation by [tex]\( -1 \)[/tex]:
[tex]\[
\frac{2}{3c} = -6
\][/tex]
Next, solve for [tex]\( c \)[/tex]:
[tex]\[
2 = -18c
\][/tex]
[tex]\[
c = \frac{-2}{18}
\][/tex]
[tex]\[
c = -\frac{1}{9}
\][/tex]
So, the value of [tex]\( c \)[/tex] is:
[tex]\[
\boxed{-\frac{1}{9}}
\][/tex]
