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The image contains a mathematical problem involving three lines labeled \( g_1 \), \( g_2 \), and \( g_3 \). We are asked to find the coordinates of points \( A \) and \( B \), with the given information that \( g_2 \) and \( g_3 \) are parallel lines. The equations of the lines and specific information provided in the problem can be broken down as follows:

### Given Information:
- Equation of \( g_1 \): \( 3x - y = 18 \)
- Equation of \( g_2 \): \( y = x + 2 \)
- Equation of \( g_3 \): not given, but it is stated that \( g_3 \parallel g_2 \) (meaning they have the same slope, and \( g_3 \) will have the form \( y = x + c \) where \( c \) is some constant).

### Task:
1. Determine the coordinates of point \( A \), where lines \( g_2 \) and \( g_3 \) intersect.
2. Determine the coordinates of point \( B \), where lines \( g_2 \) and \( g_1 \) intersect.

### Step-by-Step Solution:

#### 1. Find Point A:
Since \( g_2 \) and \( g_3 \) are parallel lines, they will never intersect. However, based on the diagram, point \( A \) likely represents the point of intersection between \( g_3 \) and the y-axis or between two other lines (possibly given more information).

#### 2. Find Point B:
To find the coordinates of point \( B \), we need to solve the system of equations for \( g_2 \) and \( g_1 \). This means solving the equations \( y = x + 2 \) and \( 3x - y = 18 \) simultaneously.

Substitute \( y = x + 2 \) into \( 3x - y = 18 \):

\[
3x - (x + 2) = 18
\]
\[
3x - x - 2 = 18
\]
\[
2x - 2 = 18
\]
\[
2x = 20
\]
\[
x = 10
\]

Now, substitute \( x = 10 \) back into \( y = x + 2 \):

\[
y = 10 + 2 = 12
\]

Thus, point \( B \) is at \( (10, 12) \).

### Additional Information:
- To determine point \( A \), we may need more clarification or assumptions about the setup, as it’s not clear from the provided information whether it's a point of intersection with another specific line.

Would you like more detailed assistance on finding point A or additional clarifications?

### Further Questions:
1. What is the role of the parallel condition for lines \( g_2 \) and \( g_3 \)?
2. How can we solve for the equation of \( g_3 \) using its parallel condition with \( g_2 \)?
3. How can we verify the points of intersection using graphing techniques?
4. Could the slope and intercepts of \( g_2 \) and \( g_1 \) lead to additional insights?
5. What does the diagram imply about the relationship between the y-values at the intersection points?

### Tip:
When solving systems of equations, always double-check substitutions to ensure correct solutions.

Determine points A and B. Given that lines g_2 and g_3 are parallel.

This problem involves solving for the coordinates of points A and B where lines intersect and are parallel. Learn how to find these points using algebraic methods and the properties of parallel lines. Suitable for students in Grades 10-11.

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