General form of line is 3x+2y-5=0, then what is the slope of the line, x intercept, y intercept, root of the equation x and y and point slope form.

Given the general form of the line \( 3x + 2y - 5 = 0 \), we can determine various properties and forms of the line:

### 1. Slope of the Line

To find the slope of the line, we convert the equation to the slope-intercept form \( y = mx + b \), where \( m \) is the slope.

Starting with:
\[ 3x + 2y - 5 = 0 \]

Solve for \( y \):
\[ 2y = -3x + 5 \]
\[ y = -\frac{3}{2}x + \frac{5}{2} \]

So, the slope \( m \) is:
\[ m = -\frac{3}{2} \]

### 2. X-Intercept

The x-intercept is the point where the line crosses the x-axis (\( y = 0 \)).

Set \( y = 0 \) in the equation \( 3x + 2y - 5 = 0 \):
\[ 3x - 5 = 0 \]
\[ 3x = 5 \]
\[ x = \frac{5}{3} \]

So, the x-intercept is:
\[ \left( \frac{5}{3}, 0 \right) \]

### 3. Y-Intercept

The y-intercept is the point where the line crosses the y-axis (\( x = 0 \)).

Set \( x = 0 \) in the equation \( 3x + 2y - 5 = 0 \):
\[ 2y - 5 = 0 \]
\[ 2y = 5 \]
\[ y = \frac{5}{2} \]

So, the y-intercept is:
\[ \left( 0, \frac{5}{2} \right) \]

### 4. Roots of the Equation

The roots of the equation are the points where the line intersects the x and y axes. These have already been found:
- x-intercept: \( \left( \frac{5}{3}, 0 \right) \)
- y-intercept: \( \left( 0, \frac{5}{2} \right) \)

### 5. Point-Slope Form

The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]

Using the slope \( m = -\frac{3}{2} \) and the y-intercept point \( (0, \frac{5}{2}) \):
\[ y - \frac{5}{2} = -\frac{3}{2}(x - 0) \]
\[ y - \frac{5}{2} = -\frac{3}{2}x \]

So, the point-slope form is:
\[ y - \frac{5}{2} = -\frac{3}{2}x \]

### Summary

1. **Slope**: \(-\frac{3}{2}\)
2. **x-intercept**: \( \left( \frac{5}{3}, 0 \right) \)
3. **y-intercept**: \( \left( 0, \frac{5}{2} \right) \)
4. **Roots**: The x-intercept and y-intercept are the roots.
5. **Point-Slope Form**: \( y - \frac{5}{2} = -\frac{3}{2}x \)

Would you like details on any part of the solution or have any other questions? Here are some related questions you might consider:

1. How do you convert the general form of a line to the standard form?
2. What is the distance from a point to a line?
3. How to find the perpendicular distance from the origin to the given line?
4. How to find the angle between two lines given their slopes?
5. How do you determine if two lines are parallel or perpendicular?

**Tip:** When converting between different forms of linear equations, always isolate \( y \) to find the slope and intercepts easily.

Explore the properties of the linear equation 3x + 2y - 5 = 0, including its slope, x-intercept, y-intercept, roots, and point-slope form. Learn step-by-step how to find these properties with detailed explanations.

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