Question

which linear function represents the line given by the point - slope equation $y + 7 = -\frac{2}{3}(x + 6)$?
$\bigcirc$ $f(x)=-\frac{2}{3}x - 11$
$\bigcirc$ $f(x)=-\frac{2}{3}x - 1$
$\bigcirc$ $f(x)=-\frac{2}{3}x + 3$
$\bigcirc$ $f(x)=-\frac{2}{3}x + 13$

Explanation:

Step1: Expand the point - slope form

We start with the point - slope equation \(y + 7=-\frac{2}{3}(x + 6)\). Using the distributive property \(a(b + c)=ab+ac\), where \(a =-\frac{2}{3}\), \(b=x\) and \(c = 6\), we get \(y+7=-\frac{2}{3}x-\frac{2}{3}\times6\).
Calculating \(-\frac{2}{3}\times6\), we have \(-\frac{2}{3}\times6=- 4\). So the equation becomes \(y + 7=-\frac{2}{3}x-4\).

Step2: Solve for y (convert to slope - intercept form)

Subtract 7 from both sides of the equation \(y+7=-\frac{2}{3}x - 4\).
\(y=-\frac{2}{3}x-4 - 7\).
Combining like terms \(-4-7=-11\), we get \(y =-\frac{2}{3}x-11\). Since \(y = f(x)\), the linear function is \(f(x)=-\frac{2}{3}x - 11\).

Answer:

\(f(x)=-\frac{2}{3}x - 11\) (the first option)