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To find the coordinates of point \(E\) where line \(OB\) intersects line \(AC\), we first need to find the equations of lines \(OB\) and \(AC\).

1. **Equation of line \(OB\):**
   Points \(O(0, 0)\) and \(B(-3, 4)\).

   The slope \(m\) of \(OB\) is:
   \[
   m_{OB} = \frac{4 - 0}{-3 - 0} = -\frac{4}{3}
   \]

   Using the point-slope form of a line equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (0, 0)\):
   \[
   y = -\frac{4}{3}x
   \]

2. **Equation of line \(AC\):**
   Points \(A(2, 0)\) and \(C(-3, 2)\).

   The slope \(m\) of \(AC\) is:
   \[
   m_{AC} = \frac{2 - 0}{-3 - 2} = -\frac{2}{5}
   \]

   Using the point-slope form of a line equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (2, 0)\):
   \[
   y - 0 = -\frac{2}{5}(x - 2) \\
   y = -\frac{2}{5}x + \frac{4}{5}
   \]

3. **Finding the intersection point \(E\):**
   Set the equations equal to each other:
   \[
   -\frac{4}{3}x = -\frac{2}{5}x + \frac{4}{5}
   \]

   Solving for \(x\):
   \[
   -\frac{4}{3}x + \frac{2}{5}x = \frac{4}{5} \\
   \left(-\frac{4}{3} + \frac{2}{5}\right)x = \frac{4}{5} \\
   \left(-\frac{20}{15} + \frac{6}{15}\right)x = \frac{4}{5} \\
   -\frac{14}{15}x = \frac{4}{5} \\
   x = \frac{4}{5} \times -\frac{15}{14} \\
   x = -\frac{12}{14} = -\frac{6}{7}
   \]

   Substituting \(x = -\frac{6}{7}\) into the equation of \(OB\):
   \[
   y = -\frac{4}{3}\left(-\frac{6}{7}\right) \\
   y = \frac{24}{21} = \frac{8}{7}
   \]

Therefore, the coordinates of point \(E\) are:
\[
E\left(-\frac{6}{7}, \frac{8}{7}\right)
\]

Would you like further details or have any questions about this solution?

Here are five questions you might ask next:
1. How do you find the equation of a line given two points?
2. What is the significance of the slope in the equation of a line?
3. How do you solve a system of linear equations?
4. Can you explain the point-slope form of a line equation?
5. How do you find the intersection of two lines algebraically?

**Tip:** Always check your calculations by substituting the solution back into the original equations to ensure they hold true.

Learn how to find the coordinates of intersection point E where lines OB and AC intersect using analytic geometry. This problem involves calculating slopes, using point-slope form, and solving a system of equations. Suitable for high school students.

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