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.