To find the slope of the line given by the equation [tex]\(2x - 6y = 9\)[/tex], we should first rearrange the equation into the slope-intercept form, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here are the steps:
1. Start with the given equation:
[tex]\[
2x - 6y = 9
\][/tex]
2. Isolate [tex]\( y \)[/tex] by moving all terms involving [tex]\( y \)[/tex] to one side of the equation:
[tex]\[
-6y = -2x + 9
\][/tex]
3. Divide every term in the equation by [tex]\(-6\)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-2x + 9}{-6}
\][/tex]
4. Simplify the fraction for each term:
[tex]\[
y = \frac{-2}{-6}x + \frac{9}{-6}
\][/tex]
5. Further simplify the fractions:
[tex]\[
y = \frac{1}{3}x - \frac{3}{2}
\][/tex]
Now the equation is in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex]. Therefore, the slope [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{1}{3}
\][/tex]
Hence, the slope of the line [tex]\(2x - 6y = 9\)[/tex] is:
[tex]\[
\frac{1}{3} \approx 0.3333
\][/tex]
