To solve this problem, we need to analyze both the systems of equations and their corresponding graphs. I will carefully match each system with its graph and provide the correct sequence.

Let me work through this now.Here are the steps to solve this problem:

Step 1: Analyze the systems of equations

1. $\{y = \sqrt{x}, \, y = x^2\}$

• $y = \sqrt{x}$: A curve passing through $(0,0)$ and rising in the first quadrant only.
• $y = x^2$: A parabola opening upwards.

2. $\{y = x^2, \, y = |x|\}$

• $y = x^2$: Again, a parabola opening upwards.
• $y = |x|$: A V-shaped graph symmetric about the $y$-axis.

3. $\{y = \frac{2}{x}, \, y = x - 1\}$

• $y = \frac{2}{x}$: A hyperbola with two branches.
• $y = x - 1$: A straight line passing through $(1,0)$.

4. $\{y = \sqrt{x}, \, y = 2x - 6\}$

• $y = \sqrt{x}$: A curve in the first quadrant only.
• $y = 2x - 6$: A straight line with slope 2 passing through $(3,0)$.

Step 2: Match graphs

Let’s compare these equations with the provided graphs:

1. Graph 1: Contains a parabola and a curve in the first quadrant (matches $y = x^2$ and $y = \sqrt{x}$).
2. Graph 2: Contains a parabola and a V-shaped graph (matches $y = x^2$ and $y = |x|$).
3. Graph 3: Contains a hyperbola and a straight line (matches $y = \frac{2}{x}$ and $y = x - 1$).
4. Graph 4: Contains a curve in the first quadrant and a straight line (matches $y = \sqrt{x}$ and $y = 2x - 6$).

Step 3: Solution sequence

The correct sequence is 1234.

Do you need further explanation or details on any of these matches? 😊

Here are 5 related questions to explore further:

1. How do you graph $y = \sqrt{x}$ step by step?
2. What are the properties of a hyperbola, such as $y = \frac{2}{x}$?
3. Why does $y = |x|$ form a V-shape?
4. How can we find the intersection points of $y = 2x - 6$ and $y = \sqrt{x}$?
5. What are some real-world applications of systems of equations involving these graphs?

Tip: For matching equations to graphs, focus on the basic shapes and key features like symmetry and asymptotes.