Let's examine the system of equations in the image:

$x + y = 4$ $x - y = 10$

We can solve this system using the elimination method.

Step 1: Add the Equations

Add the two equations together to eliminate $y$:

$(x + y) + (x - y) = 4 + 10$ $2x = 14$

Now, divide by 2 to solve for $x$:

$x = 7$

Step 2: Substitute $x = 7$ into One of the Equations

Now that we know $x = 7$, substitute it back into the first equation to find $y$:

$7 + y = 4$ Subtract 7 from both sides:

$y = -3$

Solution

The solution to the system is: $(x, y) = (7, -3)$

Consistency and Independence

Since we found a unique solution, the system is consistent (it has at least one solution), and the equations are independent (they are not multiples of each other).

Thus, the correct answer choices are:

• A: "There is one solution. The solution of the system is (7, -3)."
• C: "The system is consistent and the equations are independent."

Would you like more details or have any questions?

Here are some questions to explore:

1. What would happen if we had added the equations and found no solution?
2. How can we verify if two equations are dependent without solving them?
3. What does it mean for a system to be inconsistent?
4. Can we solve this system graphically and get the same solution?
5. How does elimination differ from substitution in solving systems?

Tip: When solving by elimination, always check if adding or subtracting equations will simplify the system effectively.