Sure! Let's convert the point-slope form equation [tex]\( y + 7 = \frac{2}{3}(x + 6) \)[/tex] into the slope-intercept form [tex]\( y = mx + b \)[/tex].
### Step-by-step solution:
1. Starting Equation:
[tex]\[
y + 7 = \frac{2}{3}(x + 6)
\][/tex]
2. Distribute [tex]\(\frac{2}{3}\)[/tex] through the terms inside the parentheses:
[tex]\[
y + 7 = \frac{2}{3}x + \frac{2}{3} \cdot 6
\][/tex]
3. Multiply [tex]\(\frac{2}{3}\)[/tex] by 6:
[tex]\[
y + 7 = \frac{2}{3}x + 4
\][/tex]
4. Isolate [tex]\( y \)[/tex] by subtracting 7 from both sides:
[tex]\[
y = \frac{2}{3}x + 4 - 7
\][/tex]
5. Simplify the constants:
[tex]\[
y = \frac{2}{3}x - 3
\][/tex]
So, the slope-intercept form of the line is [tex]\( y = \frac{2}{3}x - 3 \)[/tex].
None of the options provided exactly match this equation, but the correct transformation of the given point-slope form is:
[tex]\[
y = \frac{2}{3}x - 3
\][/tex]
Therefore, the correct linear function that represents the line given by the point-slope equation [tex]\( y + 7 = \frac{2}{3}(x + 6) \)[/tex] is [tex]\( y = \frac{2}{3}x - 3 \)[/tex].
