Question

If \(\begin{bmatrix}3& 2x + 5y &-2 \\x + 4y  &7 &-5\end{bmatrix}=\begin{bmatrix}3 &10&-2\\ 2&7&-5\end{bmatrix}\) Then the values of x and y are:

• A. x=-2, y = 6
• B. x=12, y=5
• C. x=-4, y=7
• D. x=10, y = -2

Answer 1

The Correct Option is D

The given matrix equation is:\[\begin{bmatrix}3 & 2x + 5y & -2 \\ x + 4y & 7 & -5\end{bmatrix} = \begin{bmatrix}3 & 10 & -2 \\ 2 & 7 & -5\end{bmatrix}\] We equate corresponding elements from both matrices:

1. The element in the first column, first row is identical in both matrices:\(3 = 3\), thus no new information is derived.
2. For the first row, second column, equate:\(2x + 5y = 10\)
3. For the second row, first column, equate:\(x + 4y = 2\)

We now have a system of linear equations:

1. \(2x + 5y = 10\)
2. \(x + 4y = 2\)

To solve this system, we can use substitution or elimination. We'll use the elimination method here:

1. Multiply the second equation by 2:\[2(x + 4y) = 2 \cdot 2 \Rightarrow 2x + 8y = 4\]
2. Subtract the first equation from this new equation:\[(2x + 8y) - (2x + 5y) = 4 - 10 \Rightarrow 3y = -6\]
3. Solve for y by dividing both sides by 3:\(y = -2\)
4. Substitute y back into the second original equation:\(x + 4(-2) = 2\)
5. Solve for x:\(x - 8 = 2 \Rightarrow x = 10\)

The solution is x = 10 and y = -2.