Question

which of the following equations could correspond to the network diagram above (select all that apply)

\\(x_1 - x_2 + 125 = 0\\)
\\(x_1 - x_2 = 125\\)
\\(x_1 - x_2 = -125\\)
\\(x_1 + x_2 = 125\\)
\\(x_1 + x_2 - 125 = 0\\)
\\(x_1 + 125 = x_2\\)

Explanation:

Analyze the network junction flow

To find the correct equations, we apply the principle of flow conservation at the junction (node). The total flow entering the node must equal the total flow leaving the node.

Looking at the arrows in the diagram:

• The arrow for \(x_1\) points toward the node (incoming flow).
• The arrow for \(125\) points toward the node (incoming flow).
• The arrow for \(x_2\) points away from the node (outgoing flow).

Therefore, the conservation equation is:

$$ x_1 + 125 = x_2 $$

Rearrange the conservation equation

Using the General Solution of Linear Systems knowledge point, we can rewrite this linear equation in equivalent algebraic forms:

1. Subtract \(x_2\) from both sides:

$$ x_1 - x_2 + 125 = 0 $$

1. Subtract \(125\) from both sides:

$$ x_1 - x_2 = -125 $$

Match with the given options

Let's evaluate each of the provided choices against our derived equivalent equations:

• \(x_1 - x_2 + 125 = 0\): This is equivalent to our first rearranged form. (Correct)
• \(x_1 - x_2 = 125\): Incorrect, the sign of \(125\) is wrong.
• \(x_1 - x_2 = -125\): This is equivalent to our second rearranged form. (Correct)
• \(x_1 + x_2 = 125\): Incorrect.
• \(x_1 + x_2 - 125 = 0\): Incorrect.
• \(x_1 + 125 = x_2\): This is our direct conservation equation. (Correct)

Answer:

• (A) \(x_1 - x_2 + 125 = 0\) (Correct answer)
• (B) \(x_1 - x_2 = 125\)
• (C) \(x_1 - x_2 = -125\) (Correct answer)
• (D) \(x_1 + x_2 = 125\)
• (E) \(x_1 + x_2 - 125 = 0\)
• (F) \(x_1 + 125 = x_2\) (Correct answer)