what is the equation of the line parellel to 2x - 3y =9 that also passes through the point ( 6, -1 )

To find the equation of a line parallel to \(2x - 3y = 9\) that passes through the point \( (6, -1) \), we need to follow these steps:

### Step 1: Rewrite the original line in slope-intercept form
The equation of the original line is:
\[
2x - 3y = 9
\]
To find the slope, rewrite it in the form \( y = mx + b \), where \( m \) is the slope.

1. Subtract \( 2x \) from both sides:
   \[
   -3y = -2x + 9
   \]
2. Divide by \(-3\):
   \[
   y = \frac{2}{3}x - 3
   \]

So, the slope \( m \) of the original line is \( \frac{2}{3} \).

### Step 2: Use the point-slope form for the parallel line
Since parallel lines have the same slope, the slope of our new line will also be \( \frac{2}{3} \). We’ll use the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) = (6, -1) \) and \( m = \frac{2}{3} \).

Substitute these values:
\[
y - (-1) = \frac{2}{3}(x - 6)
\]
\[
y + 1 = \frac{2}{3}x - 4
\]

### Step 3: Simplify to slope-intercept form
Subtract 1 from both sides:
\[
y = \frac{2}{3}x - 5
\]

### Final Answer
The equation of the line parallel to \( 2x - 3y = 9 \) that passes through \( (6, -1) \) is:
\[
y = \frac{2}{3}x - 5
\]

Would you like further details, or do you have any questions?

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Here are five related questions for further exploration:
1. How do we find the slope of a line given two points?
2. How do we find the equation of a line perpendicular to a given line?
3. What is the significance of parallel and perpendicular slopes?
4. How can we verify if two lines are parallel?
5. How do we convert an equation in point-slope form to standard form?

**Tip:** Remember, parallel lines always have identical slopes!

What is the equation of the line parallel to 2x - 3y = 9 that also passes through the point (6, -1)?

Discover how to find the equation of a line parallel to 2x - 3y = 9 that passes through the point (6, -1). This solution explores key algebraic concepts, including slope-intercept and point-slope forms, and is ideal for students in Grades 9-10.

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