Solve the pairs of simultaneous equatic a) \( x+y=5 ; \quad x-y=3 \) b) \( x-3 y=-5 ; \quad 2 x+3 y=-1 \) c) \( 2 y-3 x=-21 ; \quad 2 x+3 y=-1 \) d) \( 3 x+2 y=60 ; \quad 3 x-3 y=45 \) e) \( x+y=36 ; \quad x-2 y=-12 \) f) \( x+2 y=5 ; \quad 3 x-y \)

To solve the pairs of simultaneous equations, let's tackle each one step by step. a) 1. From \(x + y = 5\), we can express \(y = 5 - x\). 2. Substitute \(y\) in the second equation: \(x - (5 - x) = 3 \Rightarrow 2x - 5 = 3 \Rightarrow 2x = 8 \Rightarrow x = 4\). 3. Substitute \(x = 4\) into \(y = 5 - x\): \(y = 5 - 4 = 1\). Solution: \( (x, y) = (4, 1) \). b) 1. Rearranging \(x - 3y = -5\) gives \(x = 3y - 5\). 2. Substitute into the second equation: \(2(3y - 5) + 3y = -1 \Rightarrow 6y - 10 + 3y = -1\). 3. Combine like terms: \(9y - 10 = -1 \Rightarrow 9y = 9 \Rightarrow y = 1\). 4. Substitute \(y = 1\) into \(x = 3y - 5\): \(x = 3(1) - 5 = -2\). Solution: \( (x, y) = (-2, 1) \). c) 1. From \(2y - 3x = -21\), express \(y\): \(y = \frac{3x - 21}{2}\). 2. Substitute into \(2x + 3y = -1\): \(2x + 3\left(\frac{3x - 21}{2}\right) = -1\). 3. Multiply through by 2 to eliminate the fraction: \(4x + 9x - 63 = -2 \Rightarrow 13x = 61 \Rightarrow x = \frac{61}{13}\). 4. Substitute \(x\) back into \(y\) expression: \(y = \frac{3\left(\frac{61}{13}\right) - 21}{2}\). Solution: \( (x, y) \) is a bit complex; you can compute. d) 1. Use the first equation \(3x + 2y = 60\) to express \(y = \frac{60 - 3x}{2}\). 2. Substitute into the second equation: \(3x - 3\left(\frac{60 - 3x}{2}\right) = 45\). 3. Continue to solve: \(3x - \frac{180 - 9x}{2} = 45\). Solution: Solve for \(x\) and find \(y\). e) 1. Solve \(x + y = 36\) for \(y\): \(y = 36 - x\). 2. Substitute in \(x - 2(36 - x) = -12\): \(x - 72 + 2x = -12 \Rightarrow 3x = 60 \Rightarrow x = 20\). 3. Substitute back to find \(y\): \(y = 36 - 20 = 16\). Solution: \( (x, y) = (20, 16) \). f) 1. From \(x + 2y = 5\), express \(x\): \(x = 5 - 2y\). 2. Substitute \(x\) back into the equation (assuming it was cut off): If the intention was \(3x - y = ?\), specify an equality to proceed. Solution: Requires clarification on the second equation. Feel free to ask if you want to dive deeper into any of these solutions or if you need help with another set of equations!