Question

Find the equation of the line parallel to.
y, equals, start fraction, 2, divided by, 3, end fraction, x, plus, 1, .y=
3
2
​
x+1
and going through the point. , left parenthesis, 6, comma, minus, 5, right parenthesis, .
(6,−5)

your answer should be in slope intercept form. , y, equals, m, x, plus, b
y=mx+b
Answer
Att

Answer 1

To find the equation of the line that is parallel to the line given by the equation \( y = \frac{2}{3}x + 1 \) and passes through the point \( (6, -5) \), we need to follow these steps:

1. **Identify the slope** of the given line:
The slope \( m \) of the line \( y = \frac{2}{3}x + 1 \) is \( \frac{2}{3} \).

2. **Use the same slope for the parallel line**:
Since the two lines are parallel, the new line will also have a slope of \( \frac{2}{3} \).

3. **Use the point-slope form of the equation of a line**:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope.

Substituting in \( (x_1, y_1) = (6, -5) \) and \( m = \frac{2}{3} \):
\[
y - (-5) = \frac{2}{3}(x - 6)
\]

4. **Simplify and convert to slope-intercept form**:
\[
y + 5 = \frac{2}{3}(x - 6)
\]
Distributing the slope:
\[
y + 5 = \frac{2}{3}x - \frac{12}{3}
\]
\[
y + 5 = \frac{2}{3}x - 4
\]
Now, subtract 5 from both sides:
\[
y = \frac{2}{3}x - 4 - 5
\]
\[
y = \frac{2}{3}x - 9
\]

Thus, the equation of the line that is parallel to the given line and passes through the point \( (6, -5) \) is

\[
\boxed{y = \frac{2}{3}x - 9}
\]