Question

Solve the following pair of equations algebraically: \[ \begin{aligned} 101x + 102y &= 304 \\ 102x + 101y &= 305 \end{aligned} \]

Answer 1

We solve the system using elimination: Add the two equations: \[ (101x + 102y) + (102x + 101y) = 304 + 305 \] \[ (101x + 102x) + (102y + 101y) = 609 \] \[ 203x + 203y = 609 \Rightarrow x + y = 3 \tag{1} \] Now subtract the second equation from the first: \[ (101x + 102y) - (102x + 101y) = 304 - 305 \] \[ (101x - 102x) + (102y - 101y) = -1 \] \[ -1x + 1y = -1 \Rightarrow y - x = -1 \tag{2} \] Solve equations (1) and (2): From (1): \(x + y = 3\) From (2): \(y - x = -1\) Add (1) and (2): \[ (x + y) + (y - x) = 3 + (-1) \Rightarrow 2y = 2 \Rightarrow y = 1 \] Substitute into (1): \(x + 1 = 3 \Rightarrow x = 2\) \[ \boxed{x = 2,\quad y = 1} \]