Question: Which of the following equations represents a line that is parallel to $y = 5x - 4$
and passes through the point, $(3, 4)$?
A. $y = 5x + 19$
B. $y = 5x - 11$
C. $y = 5x - 7$
D. $y = -\frac{1}{5} x - 4$
1. **State the problem:** We need to find the equation of a line parallel to $y = 5x - 4$ that passes through the point $(3, 4)$.
2. **Recall the rule for parallel lines:** Lines are parallel if they have the same slope.
3. The slope of the given line $y = 5x - 4$ is $5$.
4. So, the line we want must have slope $m = 5$.
5. Use the point-slope form of a line equation: $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1) = (3, 4)$ and $m = 5$.
6. Substitute values: $$y - 4 = 5(x - 3)$$
7. Simplify the right side: $$y - 4 = 5x - 15$$
8. Add $4$ to both sides: $$y = 5x - 15 + 4$$
9. Simplify: $$y = 5x - 11$$
10. Check which option matches $y = 5x - 11$.
11. Option B is $y = 5x - 11$, so the answer is **B**.