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 \(y+7=-\frac{2}{3}x-4\).

Step2: Solve for y

Subtract 7 from both sides of the equation \(y + 7=-\frac{2}{3}x-4\). So \(y=-\frac{2}{3}x-4 - 7\).
Simplify the right - hand side: \(-4-7=-11\), so \(y =-\frac{2}{3}x-11\). Since \(f(x)=y\), we have \(f(x)=-\frac{2}{3}x - 11\).

Answer:

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