To find the equation of a line that is parallel to [tex]\( y = \frac{3}{7}x + 11 \)[/tex] and passes through the point [tex]\( (-21, 42) \)[/tex], follow these steps:
1. Determine the Slope:
- Lines that are parallel to each other have the same slope.
- The slope of the given line [tex]\( y = \frac{3}{7} x + 11 \)[/tex] is [tex]\( \frac{3}{7} \)[/tex].
2. Use the Point-Slope Form:
- The point-slope form of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
- Here, [tex]\( m = \frac{3}{7} \)[/tex] and the point is [tex]\( (-21, 42) \)[/tex].
3. Substitute the Given Point and Slope into the Point-Slope Form:
[tex]\[
y - 42 = \frac{3}{7}(x + 21)
\][/tex]
4. Simplify the Equation:
[tex]\[
y - 42 = \frac{3}{7} x + \frac{3}{7} \times 21
\][/tex]
[tex]\[
y - 42 = \frac{3}{7} x + 9
\][/tex]
[tex]\[
y = \frac{3}{7} x + 9 + 42
\][/tex]
[tex]\[
y = \frac{3}{7} x + 51
\][/tex]
5. Identify the Correct Option:
- We derived the equation [tex]\( y = \frac{3}{7} x + 51 \)[/tex].
- Comparing this with the given options, the correct answer that matches is option D: [tex]\( y = \frac{3}{7} x + 51 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
