https://file.aiquickdraw.com/mathbot/file/17296055857054u4vtn2w.jpg

Let's break down this problem:

### Given:
- Line \( l \) passes through the points \( (5, 1) \) and \( (0, 3) \).
- The point \( M \) is at \( (-4, -3) \), and we need to find another point on a line parallel to \( l \) that passes through \( M \), and has integer coordinates.

### Step 1: Find the slope of line \( l \)
The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points \( (5, 1) \) and \( (0, 3) \), the slope of line \( l \) is:
\[
m = \frac{1 - 3}{5 - 0} = \frac{-2}{5}
\]

### Step 2: Find the equation of the line parallel to \( l \) and passing through \( M(-4, -3) \)
A line parallel to \( l \) will have the same slope, \( m = -\frac{2}{5} \). The equation of a line with slope \( m \) passing through point \( (x_1, y_1) \) is:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = -\frac{2}{5} \), \( x_1 = -4 \), and \( y_1 = -3 \), we get:
\[
y + 3 = -\frac{2}{5}(x + 4)
\]
Simplifying:
\[
y + 3 = -\frac{2}{5}x - \frac{8}{5}
\]
\[
y = -\frac{2}{5}x - \frac{8}{5} - 3
\]
\[
y = -\frac{2}{5}x - \frac{23}{5}
\]

### Step 3: Find a point with integer coordinates
To find integer coordinates, we substitute the given answer choices into the equation of the parallel line:
1. For \( (-2, 2) \):
   \[
   y = -\frac{2}{5}(-2) - \frac{23}{5} = \frac{4}{5} - \frac{23}{5} = -\frac{19}{5}
   \]
   Not an integer.
   
2. For \( (-9, -1) \):
   \[
   y = -\frac{2}{5}(-9) - \frac{23}{5} = \frac{18}{5} - \frac{23}{5} = -\frac{5}{5} = -1
   \]
   This works since \( y = -1 \).

Thus, the point \( (-9, -1) \) lies on the line parallel to \( l \) and passes through \( M \).

### Final Answer:
The correct point is \( (-9, -1) \).

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

Here are 5 related questions to consider:
1. How would you find the equation of the line \( l \) in slope-intercept form?
2. What is the significance of finding the slope of a line for parallel lines?
3. How can we determine if two lines are perpendicular based on their slopes?
4. What is the general form of the equation of a line, and how is it derived from the slope-intercept form?
5. How could you check if a point lies on a given line using its equation?

**Tip:** Parallel lines always have the same slope but different y-intercepts!

The following graph contains line l and point M. Line l contains points (5, 1) and (0, 3). Point M is located at (-4, -3). Find another point with integral coordinates that lies on a line parallel to line l and passes through point M.

Learn how to determine a line parallel to a given line that passes through a specific point. This problem involves understanding coordinate geometry, calculating slopes, and using the point-slope form to find lines. A detailed example with integer coordinate solutions is included. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation