https://r2.erweima.ai/mathbot/image/171924515423100eiegyg.png

Let's solve the given equations step-by-step.

1. **First Equation**:
   \[
   \frac{1}{2}(2y - 4x - 19) = 0
   \]
   Simplify the equation:
   \[
   2y - 4x - 19 = 0
   \]
   Rearrange to get \( y \) in terms of \( x \):
   \[
   2y = 4x + 19
   \]
   \[
   y = 2x + \frac{19}{2}
   \]
   So, the first equation in standard form is:
   \[
   y = 2x + 9.5
   \]

2. **Second Equation**:
   \[
   y = \frac{x}{2} - 1
   \]

3. **Plotting these equations on the graph**:
   - **Equation 1**: \( y = 2x + 9.5 \) (cyan line)
   - **Equation 2**: \( y = \frac{x}{2} - 1 \) (purple line)

   From the graph provided, we need to identify the points of intersection for these two lines to find the solution.

4. **Graph Interpretation**:
   - The equation \( y - 2x = 9.5 \) is given in the graph as the cyan line.
   - The equation \( y = \frac{x}{2} - 1 \) is a line with a slope of 0.5 and y-intercept -1.

   To find the exact point of intersection algebraically, we set the equations equal to each other:
   \[
   2x + 9.5 = \frac{x}{2} - 1
   \]
   Solve for \( x \):
   \[
   2x + 9.5 = \frac{x}{2} - 1
   \]
   \[
   4x + 19 = x - 2
   \]
   \[
   4x - x = -21
   \]
   \[
   3x = -21
   \]
   \[
   x = -7
   \]
   Substitute \( x = -7 \) into one of the equations to find \( y \):
   \[
   y = \frac{-7}{2} - 1 = -\frac{7}{2} - 1 = -\frac{9}{2}
   \]

Thus, the solution to the system of equations is:
\[
(x, y) = \left( -7, -\frac{9}{2} \right)
\]

Would you like more details on any part of this solution? Here are some related questions you can ask:

1. How do you solve a system of linear equations graphically?
2. Can you explain how to convert different forms of linear equations?
3. What are the methods to solve systems of equations algebraically?
4. How do you interpret the solution of a system of equations on a graph?
5. Can you explain the concept of slopes and intercepts in linear equations?

**Tip:** When solving systems of equations graphically, always double-check the points of intersection for accuracy.

Learn how to solve a system of linear equations using graphical and algebraic methods. This problem involves finding the intersection of two linear equations through graphing and algebraic manipulation. Suitable for students in Grades 7-9.

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