Question

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

• $f(x)=-\frac{2}{3}x - 11$
• $f(x)=-\frac{2}{3}x - 1$
• $f(x)=-\frac{2}{3}x + 3$
• $f(x)=-\frac{2}{3}x + 13$

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:
\(y+7=-\frac{2}{3}x-\frac{2}{3}\times6\)
\(y + 7=-\frac{2}{3}x-4\)

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

Subtract 7 from both sides of the equation:
\(y=-\frac{2}{3}x-4 - 7\)
\(y=-\frac{2}{3}x-11\)
Since \(f(x)=y\), the linear function is \(f(x)=-\frac{2}{3}x - 11\)

Answer:

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