To determine if \((4, 5)\) is a solution to the given system of equations, we need to substitute \(x = 4\) and \(y = 5\) into both equations and see if they hold true.
The given equations are:
- \(y = \frac{3}{4}x + 2\)
- \(y = \frac{5}{4}x + 4\)
Let's substitute \(x = 4\) into each equation:
For the first equation:
\[ y = \frac{3}{4}(4) + 2 \] \[ y = 3 + 2 \] \[ y = 5 \]
This means that \((4, 5)\) satisfies the first equation.
For the second equation:
\[ y = \frac{5}{4}(4) + 4 \] \[ y = 5 + 4 \] \[ y = 9 \]
This means that \((4, 5)\) does not satisfy the second equation.
Since \((4, 5)\) satisfies the first equation but not the second, \((4, 5)\) is not a solution to the system of equations.