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.