Question

If\(\begin{bmatrix}5x+8 & 7 \\y+3 & 10x+12 \end{bmatrix} \)=\(\begin{bmatrix}2 & 3y+1 \\5 & 0 \end{bmatrix} \)then the value of 5x + 3y is equal to:

• A. -1
• B. 8
• C. 2
• D. 0

Hint:

When solving systems of equations derived from matrix elements, break down each equation and solve them step by step. Always ensure to substitute the found values of variables back into other expressions to check consistency or perform further calculations. For problems involving matrices or system of equations, this approach helps simplify complex calculations and reach the correct solution efficiently.

Answer 1

The Correct Option is D

If two matrices are equal, then their corresponding elements must be equal. Therefore, we have the following equations from the given matrices: 

\(5x + 8 = 2\) \(...(1)\)

\(7 = 3y + 1\) \(...(2)\)

\(y + 3 = 5\) \(...(3)\)

\(10x + 12 = 0\) \(...(4)\)

From equation (1):

\(5x + 8 = 2\Rightarrow5x = 2 - 8\Rightarrow5x = -6\Rightarrow x = -\frac{6}{5}\)

From equation (3):

\(y + 3 = 5\Rightarrow y = 5 - 3\Rightarrow y = 2\)

Now, calculate \(5x + 3y\):

\(5x + 3y = 5(-\frac{6}{5}) + 3(2)\)

\(= -6 + 6 = 0\)

Thus, the value of \(5x + 3y\) is \(0\).

Answer 2

Equate corresponding elements of the matrices:

\[ 5x + 8 = 2, \quad 10x + 12 = 0, \quad y + 3 = 5, \quad 3y + 1 = 7. \]

Step 1: Solve \( 5x + 8 = 2 \):

Begin with the equation: \[ 5x + 8 = 2. \] Subtract 8 from both sides: \[ 5x = -6. \] Divide by 5: \[ x = -\frac{6}{5}. \]

Step 2: Solve \( 3y + 1 = 7 \):

Now, solve for \( y \) using the equation: \[ 3y + 1 = 7. \] Subtract 1 from both sides: \[ 3y = 6. \] Divide by 3: \[ y = 2. \]

Step 3: Calculate \( 5x + 3y \):

Substituting \( x = -\frac{6}{5} \) and \( y = 2 \) into the expression \( 5x + 3y \): \[ 5 \left( -\frac{6}{5} \right) + 3 \cdot 2 = -6 + 6 = 0. \]

Conclusion: The result of \( 5x + 3y \) is 0.