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.
When two matrices are equal, their corresponding elements must be equal. This yields the following equations from the given matrices:
\(5x + 8 = 2\) \(...(1)\)
\(7 = 3y + 1\) \(...(2)\)
\(y + 3 = 5\) \(...(3)\)
\(10x + 12 = 0\) \(...(4)\)
Solving equation (1) for x:
\(5x + 8 = 2 \Rightarrow 5x = 2 - 8 \Rightarrow 5x = -6 \Rightarrow x = -\frac{6}{5}\)
Solving equation (3) for y:
\(y + 3 = 5 \Rightarrow y = 5 - 3 \Rightarrow y = 2\)
Now, we compute the value of \(5x + 3y\):
\(5x + 3y = 5(-\frac{6}{5}) + 3(2)\)
\(= -6 + 6 = 0\)
The value of \(5x + 3y\) is \(0\).