We have the system of equations:

$1. \quad 2a + c + d = 18$ $2. \quad a + 2c + d = 16$ $3. \quad 2a + b + c + d = 24$

We need to solve for $a$, $b$, $c$, and $d$.

Step 1: Subtract the first equation from the second

$(a + 2c + d) - (2a + c + d) = 16 - 18$ Simplifying: $a + 2c + d - 2a - c - d = -2$ $-a + c = -2 \quad \Rightarrow \quad a = c + 2$

Step 2: Substitute $a = c + 2$ into the first and third equations

Substitute into Equation 1: $2(c + 2) + c + d = 18$ $2c + 4 + c + d = 18 \quad \Rightarrow \quad 3c + d = 14 \quad \text{(Equation 4)}$

Substitute into Equation 3: $2(c + 2) + b + c + d = 24$ $2c + 4 + b + c + d = 24 \quad \Rightarrow \quad 3c + b + d = 20 \quad \text{(Equation 5)}$

Step 3: Solve Equations 4 and 5

We now have the system: $4. \quad 3c + d = 14$ $5. \quad 3c + b + d = 20$

Subtract Equation 4 from Equation 5: $(3c + b + d) - (3c + d) = 20 - 14$ Simplifying: $b = 6$

Step 4: Find $d$ and $c$

Now, substitute $b = 6$ into Equation 5: $3c + 6 + d = 20 \quad \Rightarrow \quad 3c + d = 14$ This is exactly the same as Equation 4, so it confirms that $d = 14 - 3c$.

Step 5: Substitute $d = 14 - 3c$ into Equation 4

Substitute into $3c + d = 14$: $3c + (14 - 3c) = 14$ Simplifying: $3c + 14 - 3c = 14 \quad \Rightarrow \quad 14 = 14$ This is always true, so it confirms that the value of $c$ can be any number.

Step 6: Final Answer

Thus, $a = c + 2$, $b = 6$, and $d = 14 - 3c$.

Since $c$ can be any number, the system has infinitely many solutions where $a$, $b$, and $d$ depend on $c$.

Solution:

$a = c + 2, \quad b = 6, \quad d = 14 - 3c, \quad c \text{ is free}$

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Would you like to see how this works with a specific value for $c$, or do you have any further questions?

Here are some related questions for you:

1. How would you solve a system of equations with three variables and one equation missing?
2. What is the concept of free variables in a system of equations?
3. How can you represent solutions graphically in a 3D coordinate system?
4. What methods can be used for solving non-linear systems of equations?
5. How can you check the correctness of a solution to a system of equations?

Tip: When solving systems with multiple variables, keep an eye out for any dependent or free variables, as they often indicate infinite solutions!