write an equation for the line in​ point-slope form and general form.
Passing through left parenthesis negative 7 comma 1 right parenthesis
 
and parallel to the line whose equation is 2 x minus 5 y minus 3 equals 0

To write the equation of the line in both point-slope form and general form, we follow these steps:

### Step 1: Identify the slope of the given line
The given line is \( 2x - 5y - 3 = 0 \). First, we'll rewrite this equation in slope-intercept form \( y = mx + b \) to find the slope.

Start by solving for \( y \):
\[
2x - 5y - 3 = 0
\]
\[
-5y = -2x + 3
\]
\[
y = \frac{2}{5}x - \frac{3}{5}
\]

The slope \( m \) of the given line is \( \frac{2}{5} \). Since the new line is parallel to this line, it will have the same slope.

### Step 2: Use the point-slope form
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point through which the line passes. The point provided is \( (-7, 1) \) and the slope \( m = \frac{2}{5} \).

Substitute these values into the point-slope form:
\[
y - 1 = \frac{2}{5}(x + 7)
\]

This is the **point-slope form** of the equation.

### Step 3: Convert to general form
Now, let's convert the point-slope form to the general form \( Ax + By + C = 0 \).

Start by distributing the slope:
\[
y - 1 = \frac{2}{5}x + \frac{14}{5}
\]

Multiply through by 5 to eliminate the fractions:
\[
5(y - 1) = 2(x + 7)
\]
\[
5y - 5 = 2x + 14
\]

Rearrange the terms to get everything on one side:
\[
2x - 5y + 19 = 0
\]

This is the **general form** of the equation.

### Final Answer:
- Point-slope form: \( y - 1 = \frac{2}{5}(x + 7) \)
- General form: \( 2x - 5y + 19 = 0 \)

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the slope of a line perpendicular to another?
2. What is the slope-intercept form and how is it derived?
3. How do you convert a general form equation to slope-intercept form?
4. What is the relationship between parallel and perpendicular lines?
5. Can you find the equation of a line passing through two points?

**Tip:** When writing the general form of a line, ensure all coefficients are integers and the coefficient of \( x \) is positive for consistency.

write an equation for the line in point-slope form and general form. Passing through (-7, 1) and parallel to the line whose equation is 2x - 5y - 3 = 0

Learn how to find the equation of a line passing through (-7, 1) that is parallel to the line 2x - 5y - 3 = 0. Explore key concepts like point-slope form, general form, and parallel line properties, with step-by-step solutions. Ideal for Grades 9-12 students.

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