Question: A line is drawn through $(-7, 11)$ and $(8, -9)$. The equation $y - 11 = -\frac{4}{3} (x + 7)$ is written to represent the line. Which equations also represent the line? Check all that apply. y = -\frac{4}{3} x + \frac{5}{3} 3y = -4x + 40 4x + y = 21 4x + 3y = 5 -4x + 3y = 17 y - 11 = -\frac{4}{3} (x + 7)

1. **State the problem:** We are given the line equation $$y - 11 = -\frac{4}{3} (x + 7)$$ and need to find which of the given equations represent the same line. 2. **Start by distributing the right side:** $$y - 11 = -\frac{4}{3} x - \frac{4}{3} \times 7$$ $$y - 11 = -\frac{4}{3} x - \frac{28}{3}$$ 3. **Add 11 to both sides to isolate $y$:** $$y = -\frac{4}{3} x - \frac{28}{3} + 11$$ 4. **Convert 11 to thirds to combine:** $$11 = \frac{33}{3}$$ $$y = -\frac{4}{3} x - \frac{28}{3} + \frac{33}{3}$$ 5. **Combine constants:** $$y = -\frac{4}{3} x + \frac{5}{3}$$ 6. **Check each given equation:** - $y = -\frac{4}{3} x + \frac{5}{3}$ matches exactly. - $3y = -4x + 40$: Multiply both sides of our equation by 3: $$3y = 3 \left(-\frac{4}{3} x + \frac{5}{3}\right) = -4x + 5$$ This does not equal $-4x + 40$, so this is not the same line. - $4x + y = 21$: Rewrite our equation as: $$y = -\frac{4}{3} x + \frac{5}{3}$$ Multiply both sides by 3: $$3y = -4x + 5$$ Add $4x$ to both sides: $$4x + 3y = 5$$ This is not $4x + y = 21$, so no. - $4x + 3y = 5$: From above, this matches exactly our rearranged equation. - $-4x + 3y = 17$: Check if this matches: Start from $4x + 3y = 5$, multiply both sides by $-1$: $$-4x - 3y = -5$$ This is not $-4x + 3y = 17$, so no. **Final answers:** - $y = -\frac{4}{3} x + \frac{5}{3}$ - $4x + 3y = 5$ These two equations represent the same line.