Determining Sufficiency of Statements to Find Values

The question asks whether the given statements are sufficient, either individually or together, to determine the unique values of the variables x, y, and z.

We need to analyze each statement separately and then combine them to see if we can find a single, specific value for each of x, y, and z.

Analyzing Statement I: Equations with x, y, and z

Statement I provides the following two equations:

• Equation (1): $\text{x + y = 12}$
• Equation (2): $\text{x + z = 4}$

With only these two equations, we have three unknown variables (x, y, and z). A system of linear equations needs at least as many independent equations as there are variables to have a unique solution for all variables. Since we have only two equations for three variables, we cannot find unique values for x, y, and z using only Statement I. Different combinations of x, y, and z could satisfy these two equations.

For example:

• If $\text{x = 6}$, then from (1), $\text{y = 12 - 6 = 6}$, and from (2), $\text{z = 4 - 6 = -2}$. So, $\text{(x, y, z) = (6, 6, -2)}$.
• If $\text{x = 10}$, then from (1), $\text{y = 12 - 10 = 2}$, and from (2), $\text{z = 4 - 10 = -6}$. So, $\text{(x, y, z) = (10, 2, -6)}$.

Since we get different values for x, y, and z, Statement I alone is not sufficient to find unique values for all three variables.

Analyzing Statement II: Equation with x and y

Statement II provides only one equation:

• Equation (3): $\text{x - y = 6}$

With only this equation, we have two unknown variables (x and y) and no information about z. We cannot find unique values for x and y from a single linear equation with two variables. For example, if $\text{x = 10}$, $\text{y = 4}$ satisfies $\text{x - y = 6}$. If $\text{x = 7}$, $\text{y = 1}$ also satisfies $\text{x - y = 6}$. Also, there is no information about the value of z.

Therefore, Statement II alone is not sufficient to find unique values for x, y, and z.

Analyzing Statements I and II Together

Now let's consider both statements together. We have the following system of three linear equations with three variables:

1. $\text{x + y = 12}$ (From Statement I)
2. $\text{x + z = 4}$ (From Statement I)
3. $\text{x - y = 6}$ (From Statement II)

We can use equations (1) and (3) to solve for x and y, as they form a system of two equations with two variables:

• $\text{x + y = 12}$ (1)
• $\text{x - y = 6}$ (3)

Adding equation (1) and equation (3) eliminates y:

$\text{(x + y) + (x - y) = 12 + 6}$

$\text{2x = 18}$

$\text{x = }\frac{18}{2}$

$\text{x = 9}$

Now substitute the value of $\text{x = 9}$ into equation (1) to find y:

$\text{9 + y = 12}$

$\text{y = 12 - 9}$

$\text{y = 3}$

Now that we have the value of x, we can substitute $\text{x = 9}$ into equation (2) to find z:

$\text{9 + z = 4}$

$\text{z = 4 - 9}$

$\text{z = -5}$

Using both statements together, we found unique values for all three variables: $\text{x = 9}$, $\text{y = 3}$, and $\text{z = -5}$.

Conclusion on Statement Sufficiency

Statement I alone is not sufficient. Statement II alone is not sufficient. However, both statements I and II together are sufficient to determine the unique values of x, y, and z.

Revision Table: Key Concepts for Sufficiency Questions

Additional Information: Solving Systems of Linear Equations

When you have a system of linear equations, there are common methods to find the values of the variables if a unique solution exists:

• Substitution Method: Solve one equation for one variable, and then substitute that expression into the other equation(s). This reduces the number of variables and equations.
• Elimination Method: Add or subtract equations (sometimes after multiplying them by a constant) to eliminate one variable. Repeat until you have a single equation with one variable.
• Matrix Methods: For larger systems, matrices can be used (like Gaussian elimination or Cramer's rule).

In this problem, we used a combination of elimination (adding equations 1 and 3) and substitution (using the value of x in equation 2) to find the unique values of x, y, and z. The ability to find these unique values confirms that the statements combined are sufficient.