Find a line passing through intersection point of 3x - y -2 =0 and x+2y +1 =0 and -2/3 × m = -1 where m is slope of line. 

We are tasked with finding a line that passes through the intersection point of the lines given by:

1. \( 3x - y - 2 = 0 \)
2. \( x + 2y + 1 = 0 \)

And the slope \( m \) of the desired line is given by the equation \( -\frac{2}{3} \times m = -1 \).

### Step 1: Solve for the intersection point
To find the intersection point of the two lines, we need to solve the system of equations:

\[
3x - y - 2 = 0 \quad \text{(Equation 1)}
\]
\[
x + 2y + 1 = 0 \quad \text{(Equation 2)}
\]

Solve Equation 2 for \( x \):

\[
x = -2y - 1
\]

Substitute this into Equation 1:

\[
3(-2y - 1) - y - 2 = 0
\]

Expand and simplify:

\[
-6y - 3 - y - 2 = 0
\]
\[
-7y - 5 = 0
\]
\[
-7y = 5
\]
\[
y = -\frac{5}{7}
\]

Substitute \( y = -\frac{5}{7} \) back into \( x = -2y - 1 \):

\[
x = -2\left(-\frac{5}{7}\right) - 1 = \frac{10}{7} - 1 = \frac{10}{7} - \frac{7}{7} = \frac{3}{7}
\]

Thus, the intersection point is \( \left(\frac{3}{7}, -\frac{5}{7}\right) \).

### Step 2: Find the slope \( m \)
We are given the equation:

\[
-\frac{2}{3} \times m = -1
\]

Solve for \( m \):

\[
m = \frac{3}{2}
\]

### Step 3: Find the equation of the line
Now, using the point-slope form of the line equation, the line passes through \( \left(\frac{3}{7}, -\frac{5}{7}\right) \) and has slope \( m = \frac{3}{2} \).

The point-slope form is:

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

Substitute \( m = \frac{3}{2} \), \( x_1 = \frac{3}{7} \), and \( y_1 = -\frac{5}{7} \):

\[
y - \left(-\frac{5}{7}\right) = \frac{3}{2}\left(x - \frac{3}{7}\right)
\]

Simplify:

\[
y + \frac{5}{7} = \frac{3}{2}\left(x - \frac{3}{7}\right)
\]

Multiply out the right-hand side:

\[
y + \frac{5}{7} = \frac{3}{2}x - \frac{9}{14}
\]

Now subtract \( \frac{5}{7} = \frac{10}{14} \) from both sides:

\[
y = \frac{3}{2}x - \frac{9}{14} - \frac{10}{14}
\]
\[
y = \frac{3}{2}x - \frac{19}{14}
\]

Thus, the equation of the line is:

\[
y = \frac{3}{2}x - \frac{19}{14}
\]

### Summary:
The line passing through the intersection point of the two given lines and having a slope of \( \frac{3}{2} \) is:

\[
y = \frac{3}{2}x - \frac{19}{14}
\]

Would you like more details, or have any questions about this process?

### Follow-up Questions:
1. How do we solve systems of linear equations graphically?
2. What is the point-slope form of a line?
3. How does the slope of a line affect its steepness?
4. What other methods can be used to find the intersection of two lines?
5. How do we convert point-slope form to slope-intercept form?

### Tip:
Always check your solution by substituting the intersection point into the original line equations to ensure accuracy.

Find a line passing through the intersection point of 3x - y -2 = 0 and x + 2y + 1 = 0 and -2/3 × m = -1 where m is the slope of the line.

This problem involves finding a line that passes through the intersection of two linear equations, using algebraic techniques such as solving systems of equations and determining the slope of the line. The detailed solution uses the point-slope form to express the final equation. Suitable for students in Grades 9-12.

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