To find the equation of the line that goes through the point [tex]\((8, 8)\)[/tex] and is parallel to the line [tex]\(3x + 3y = 4\)[/tex], we need to determine the slope [tex]\(m\)[/tex] of the given line and then use this information to find the slope and y-intercept [tex]\(b\)[/tex] of the new line.
### Step 1: Determine the slope [tex]\(m\)[/tex] of the given line
First, we need to rewrite the equation [tex]\(3x + 3y = 4\)[/tex] in slope-intercept form ([tex]\(y = mx + b\)[/tex]) so we can easily identify the slope.
1. Divide the entire equation by 3:
[tex]\[
x + y = \frac{4}{3}
\][/tex]
2. Isolate [tex]\(y\)[/tex]:
[tex]\[
y = -x + \frac{4}{3}
\][/tex]
From this, we see that the slope [tex]\(m\)[/tex] of the given line is [tex]\(-1\)[/tex].
### Step 2: Use point-slope form to find the equation of the new line
Since the new line is parallel to the given line, it will also have the same slope [tex]\(m = -1\)[/tex].
We use the point-slope form of the line equation, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plug in the known point [tex]\((8, 8)\)[/tex] and the slope [tex]\(-1\)[/tex]:
[tex]\[
y - 8 = -1(x - 8)
\][/tex]
### Step 3: Simplify the equation to find the y-intercept [tex]\(b\)[/tex]
1. Distribute [tex]\(-1\)[/tex]:
[tex]\[
y - 8 = -x + 8
\][/tex]
2. Isolate [tex]\(y\)[/tex]:
[tex]\[
y = -x + 16
\][/tex]
So, the y-intercept [tex]\(b\)[/tex] is 16.
### Result
The equation of the line that goes through the point [tex]\((8, 8)\)[/tex] and is parallel to the line [tex]\(3x + 3y = 4\)[/tex] is:
[tex]\[
y = -x + 16
\][/tex]
Therefore:
- The slope [tex]\(m\)[/tex] is [tex]\(-1\)[/tex].
- The y-intercept [tex]\(b\)[/tex] is [tex]\(16\)[/tex].
So the answers are:
- [tex]\(m\)[/tex] is: [tex]\(\boxed{-1}\)[/tex]
- [tex]\(b\)[/tex] is: [tex]\(\boxed{16}\)[/tex]
