Write \(\displaystyle 3x+3y=1\) in slope intercept form, \(\displaystyle y=mx+b\), to determine the slope, \(\displaystyle m\):

\(\displaystyle 3y=-3x+1\)

\(\displaystyle y=-x+\frac{1}{3}\)

The slope is:

\(\displaystyle m=-1\)

Given the slope, use the point \(\displaystyle (2,2)\) and the equation \(\displaystyle y=mx+b\) to solve for the value of the \(\displaystyle y\)-intercept, \(\displaystyle b\). Substitute the known values.

\(\displaystyle 2=(-1)(2)+b\)

\(\displaystyle b=4\)

With the known slope and the \(\displaystyle y\)-intercept, plug both values back to the slope intercept formula. The answer is \(\displaystyle y=-x+4\).

The definition of a parallel line is that the lines have the same slopes, but different intercepts. The only answer with the same slope is \(\displaystyle y=3x-12\).

Equations that are parallel have the same slope.

For the equation:

\(\displaystyle y=3x+4\)

The slope is \(\displaystyle 3\) since that is how much \(\displaystyle y\) changes with increment of \(\displaystyle x\).

The only other equation with a slope of \(\displaystyle 3\) is:

\(\displaystyle y=3x+1\)

To find a parallel line to

\(\displaystyle y=x\)

we need to find another equation with the same slope of \(\displaystyle 1\) or \(\displaystyle x\).

The only equation that satisfies this is \(\displaystyle y=x+42746\).

To find an equation that is parallel to

\(\displaystyle y=5x+5\)

we need to find an equation with the same slope of \(\displaystyle 5\).

Basically we are looking for another equation with \(\displaystyle 5x\).

The only other equation that satisfies this is

\(\displaystyle y=5x+10\).

Two parallel lines have the same slope. Therefore, it is necessary to find the slope of the line of the equation 

\(\displaystyle 5x+ 7y = 32\)

Rewrite the equation in slope-intercept form \(\displaystyle y = mx+ b\). \(\displaystyle m\), the coefficient of \(\displaystyle x\), will be the slope of the line.

Add \(\displaystyle -5x\) to both sides:

\(\displaystyle 5x+ 7y + (-5x ) = 32 + (-5x )\)

\(\displaystyle 7y = -5x+32\)

Multiply both sides by \(\displaystyle \frac{1}{7}\), distributing on the right:

\(\displaystyle \frac{1}{7} \cdot 7y = \frac{1}{7} \cdot( -5x+32)\)

\(\displaystyle y = -\frac{5}{7}x+\frac{32}{7}\)

The slope of this line is \(\displaystyle -\frac{5}{7}\). The slope of the first line will be the same. The slope-intercept form of the equation of this line will be 

\(\displaystyle y =-\frac{5}{7}x+b\).

To find \(\displaystyle b\), set \(\displaystyle x = -2\) and \(\displaystyle y = 8\) and solve:

\(\displaystyle 8 =-\frac{5}{7} (-2)+b\)

\(\displaystyle 8 = \frac{10}{7} +b\)

\(\displaystyle \frac{56}{7} = \frac{10}{7} +b\)

Subtract \(\displaystyle \frac{10}{7}\) from both sides:

\(\displaystyle \frac{56}{7} - \frac{10}{7} = \frac{10}{7} +b - \frac{10}{7}\)

\(\displaystyle b = \frac{46}{7}\)

The slope-intercept form of the equation is \(\displaystyle y =-\frac{5}{7}x+ \frac{46}{7}\)

To rewrite in standard form with integer coefficients:

Multiply both sides by 7:

\(\displaystyle 7 y =7\left (-\frac{5}{7}x+ \frac{46}{7} \right )\)

\(\displaystyle 7 y =7\left (-\frac{5}{7}x \right ) +7\left ( \frac{46}{7} \right )\)

\(\displaystyle 7 y =-5x +46\)

Add \(\displaystyle 5x\) to both sides:

\(\displaystyle 7 y + 5x =-5x +46 + 5x\)

\(\displaystyle 5x+ 7y = 46\),

the correct equation in standard form.