A line passes through the points A(-3,-2) and B(2,1). Does it also pass through the point C(5,3)
? Justify your conclusion.. 

To determine if the line passing through points \( A(-3, -2) \) and \( B(2, 1) \) also passes through point \( C(5, 3) \), we can follow these steps:

### Step 1: Find the slope of the line passing through \( A \) and \( B \)
The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using points \( A(-3, -2) \) and \( B(2, 1) \):

\[
m = \frac{1 - (-2)}{2 - (-3)} = \frac{1 + 2}{2 + 3} = \frac{3}{5}
\]

So, the slope of the line is \( \frac{3}{5} \).

### Step 2: Equation of the line passing through \( A \) and \( B \)
We now use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

Using point \( A(-3, -2) \) and slope \( \frac{3}{5} \), the equation becomes:

\[
y - (-2) = \frac{3}{5}(x - (-3))
\]
\[
y + 2 = \frac{3}{5}(x + 3)
\]
\[
y + 2 = \frac{3}{5}x + \frac{9}{5}
\]
\[
y = \frac{3}{5}x + \frac{9}{5} - \frac{10}{5}
\]
\[
y = \frac{3}{5}x - \frac{1}{5}
\]

So, the equation of the line is:

\[
y = \frac{3}{5}x - \frac{1}{5}
\]

### Step 3: Check if point \( C(5, 3) \) satisfies the line equation
Substitute \( x = 5 \) and \( y = 3 \) into the equation of the line:

\[
y = \frac{3}{5}(5) - \frac{1}{5}
\]
\[
y = 3 - \frac{1}{5} = \frac{15}{5} - \frac{1}{5} = \frac{14}{5}
\]

This gives \( y = \frac{14}{5} \), which is approximately 2.8, not 3. Hence, point \( C(5, 3) \) **does not lie on the line**.

### Conclusion:
The line passing through points \( A(-3, -2) \) and \( B(2, 1) \) **does not pass through** point \( C(5, 3) \).

Would you like more details on any part of this? 

Here are some related questions:
1. How do we find the equation of a line using different forms (e.g., slope-intercept)?
2. Can we use a different method to verify if a point lies on a line?
3. What are parallel and perpendicular lines in relation to slopes?
4. How do we find the distance from a point to a line?
5. What is the significance of slope in geometry?

**Tip**: The slope of a line represents its steepness; lines with the same slope are parallel!

A line passes through the points A(-3,-2) and B(2,1). Does it also pass through the point C(5,3)? Justify your conclusion.

Find out if the line passing through points A(-3,-2) and B(2,1) also passes through point C(5,3). This problem explores coordinate geometry, slope calculation, and the equation of a line. Detailed steps are provided to justify the solution. Suitable for students in Grades 8-10.

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