A binary operation is defined by x y = xy. If x * 2 = 12 − x, find the possible values of x.

Given $x * y = xy$, we can find $x * 2$ by substituting $y=2$:

$$x * 2 = x \cdot 2 = 2x$$

We are given that $x * 2 = 12 - x$. Equating this with our result from Step 1:

$$2x = 12 - x$$

Add $x$ to both sides:

$$2x + x = 12$$ $$3x = 12$$

Divide by 3:

$$x = \frac{12}{3}$$ $$x = 4$$

Note on options: The literal interpretation of the problem yields a single value $x=4$. However, the multiple-choice options are pairs of values, suggesting a quadratic equation was intended. A common scenario for this type of question is a typo where $x*2$ was meant to be $x*x$. If we assume the question intended $x*x = 12-x$:

$$x \cdot x = 12 - x$$ $$x^2 = 12 - x$$

Rearrange into a standard quadratic form:

$$x^2 + x - 12 = 0$$

Factor the quadratic equation:

$$(x+4)(x-3) = 0$$

This gives two possible values for $x$:

$$x+4 = 0 \implies x = -4$$ $$x-3 = 0 \implies x = 3$$

The values are $3, -4$. This matches option C. Given the format of the options, this interpretation is highly probable.

The final answer is $\boxed{\text{C. } 3, -4}$.

22. Find $y$, if $\begin{pmatrix} 5 & -6 \\ 2 & -7 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ -11 \end{pmatrix}$

Multiply the $2 \times 2$ matrix by the $2 \times 1$ column vector:

$$\begin{pmatrix} (5)(x) + (-6)(y) \\ (2)(x) + (-7)(y) \end{pmatrix} = \begin{pmatrix} 7 \\ -11 \end{pmatrix}$$ $$\begin{pmatrix} 5x - 6y \\ 2x - 7y \end{pmatrix} = \begin{pmatrix} 7 \\ -11 \end{pmatrix}$$

Equating the corresponding elements of the matrices gives:

1. $5x - 6y = 7$
2. $2x - 7y = -11$

To eliminate $x$, multiply equation (1) by 2 and equation (2) by 5:

$$2 \times (5x - 6y = 7) \implies 10x - 12y = 14 \quad \text{(Equation 3)}$$ $$5 \times (2x - 7y = -11) \implies 10x - 35y = -55 \quad \text{(Equation 4)}$$