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

1. Analyze the question

We are given a matrix equation involving a $2 \times 2$ matrix multiplied by a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$, resulting in another column vector. We need to solve for the value of $y$.

2. Explain the concept

To solve this, we perform matrix multiplication. When you multiply a $2 \times 2$ matrix by a $2 \times 1$ column vector, you multiply the rows of the first matrix by the column of the second matrix to form a system of linear equations: $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$

By equating the resulting matrix to the given vector $\begin{pmatrix} 7 \\ -11 \end{pmatrix}$, we get two simultaneous equations.

3. Show the solution

Step 1: Set up the equations Multiply the matrix: Row 1: $5x - 6y = 7$ --- (Equation 1) Row 2: $2x - 7y = -11$ --- (Equation 2)

Step 2: Solve the simultaneous equations We can use the elimination method. To eliminate $x$, multiply Equation 1 by 2 and Equation 2 by 5: $(5x - 6y = 7) \times 2 \implies 10x - 12y = 14$ --- (Equation 3) $(2x - 7y = -11) \times 5 \implies 10x - 35y = -55$ --- (Equation 4)

Step 3: Subtract Equation 4 from Equation 3 $(10x - 12y) - (10x - 35y) = 14 - (-55)$ $10x - 10x - 12y + 35y = 14 + 55$ $23y = 69$

Step 4: Solve for $y$ $y = \frac{69}{23}$ $y = 3$

4. Identify the answer

Comparing our result to the options provided: A. 8 B. 5 C. 3 D. 2

The correct option is C.

5. Tip

When solving matrix equations, always check your answer by substituting $x$ and $y$ back into the original equations. If we solve for $x$ using $y=3$: $5x - 6(3) = 7 \implies 5x - 18 = 7 \implies 5x = 25 \implies x = 5$. Checking in Equation 2: $2(5) - 7(3) = 10 - 21 = -11$. It matches perfectly!