What is the equation of the line that passes through the point (6, 14) and is parallel to the line with the equation [tex]\(y = -\frac{4}{3} x - 1\)[/tex]?

A. [tex]\(y = -\frac{4}{3} x + 6\)[/tex]

B. [tex]\(y = \frac{3}{4} x + 8\)[/tex]

C. [tex]\(y = \frac{3}{4} x + 20\)[/tex]

D. [tex]\(y = -\frac{4}{3} x + 22\)[/tex]



Answer :

To find the equation of the line that passes through the point [tex]\((6,14)\)[/tex] and is parallel to the line given by the equation [tex]\(y = -\frac{4}{3}x - 1\)[/tex], follow these steps:

1. Identify the slope of the given line:
The equation of the line is [tex]\(y = -\frac{4}{3}x - 1\)[/tex]. This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Thus, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{4}{3}\)[/tex].

2. Determine the slope of the new, parallel line:
Lines that are parallel have the same slope. Therefore, the slope of the new line is also [tex]\(-\frac{4}{3}\)[/tex].

3. Use the point-slope form to find the equation of the new line:
The point-slope form of a line's equation 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. In this case, the line must pass through [tex]\((6, 14)\)[/tex] and has a slope of [tex]\(-\frac{4}{3}\)[/tex].

4. Plug in the known values:
[tex]\[ y - 14 = -\frac{4}{3}(x - 6) \][/tex]

5. Distribute the slope and simplify:
[tex]\[ y - 14 = -\frac{4}{3}x + \frac{4}{3} \cdot 6 \][/tex]
[tex]\[ y - 14 = -\frac{4}{3}x + 8 \][/tex]

6. Solve for [tex]\(y\)[/tex] (convert to slope-intercept form):
[tex]\[ y = -\frac{4}{3}x + 8 + 14 \][/tex]
[tex]\[ y = -\frac{4}{3}x + 22 \][/tex]

So, the equation of the line that passes through the point [tex]\((6,14)\)[/tex] and is parallel to the line [tex]\(y = -\frac{4}{3}x - 1\)[/tex] is:
[tex]\[ y = -\frac{4}{3}x + 22 \][/tex]

Hence, the correct answer is D. [tex]\(y = -\frac{4}{3}x + 22\)[/tex].