If the points (-3, 5), (x, y) and (-2, 3) lie on a straight line, then what is the equation of the line?

To find the equation of the line that passes through the points \((-3, 5)\), \((x, y)\), and \((-2, 3)\), follow these steps:

### 1. Calculate the slope between two points
First, calculate the slope (\(m\)) of the line using the points \((-3, 5)\) and \((-2, 3)\). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Substitute the coordinates \((x_1, y_1) = (-3, 5)\) and \((x_2, y_2) = (-2, 3)\) into the formula:

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

So, the slope \(m\) is \(1\).

### 2. Use the slope-point form to find the equation of the line
Now that we have the slope \(m = 1\), we can use the point-slope form of the equation of a line, which is:

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

Substitute the slope \(m = 1\) and the coordinates of one of the points, say \((-3, 5)\), into the equation:

\[
y - 5 = 1(x - (-3))
\]

Simplify the equation:

\[
y - 5 = x + 3
\]

\[
y = x + 8
\]

### 3. Verify the equation with the second point
We found the equation \(y = x + 8\). To verify, substitute the second point \((-2, 3)\) into this equation:

\[
3 = -2 + 8
\]

\[
3 = 6
\]

This result seems incorrect, meaning an error occurred. Let's re-evaluate the slope calculation and equation.

### 4. Re-calculate the correct slope and line equation
The previous steps revealed a possible miscalculation, so let's find the slope again between \((-3, 5)\) and \((-2, 3)\):

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

Now, using point-slope form with this corrected slope:

\[
y - 5 = -2(x + 3)
\]

Simplify the equation:

\[
y - 5 = -2x - 6
\]

\[
y = -2x - 1
\]

### 5. Verify with the second point
Substituting \((-2, 3)\) into \(y = -2x - 1\):

\[
3 = -2(-2) - 1 = 4 - 1 = 3
\]

This matches, confirming that the correct equation of the line is:

\[
\boxed{y = -2x - 1}
\]

Would you like further details or have any other questions?

Here are 5 related questions to explore further:

1. How do you determine if three points lie on the same straight line?
2. What is the general form of the equation of a line?
3. How do you find the equation of a line parallel to a given line through a specific point?
4. What is the relationship between the slopes of perpendicular lines?
5. How can you derive the equation of a line from its slope and a point?

**Tip:** Always double-check your slope calculations and use a consistent approach to verify the correctness of your linear equations.

Learn how to find the equation of a line passing through specified points (-3, 5), (x, y), and (-2, 3). This problem involves calculating the slope between points and applying the point-slope form of a line equation. Suitable for students in Grades 7-9.

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