Solve the following systems of simultaneous equations.

Here are the solutions to the problems: 1. Solve the system of the simultaneous equations i. Given the system: 3x - y = 1 &(1) \\ x - y = -3 &(2) Step 1: Subtract equation (2) from equation (1) to eliminate y. (3x - y) - (x - y) = 1 - (-3) 3x - y - x + y = 1 + 3 2x = 4 Step 2: Solve for x. x = (4)/(2) x = 2 Step 3: Substitute x=2 into equation (2) to find y. 2 - y = -3 -y = -3 - 2 -y = -5 y = 5 The solution is x=2, y=5. ii. Given the system: 3x - 4y = 0 &(1) \\ x - y = 2 &(2) Step 1: From equation (2), express x in terms of y. x = y + 2 Step 2: Substitute this expression for x into equation (1). 3(y+2) - 4y = 0 3y + 6 - 4y = 0 -y + 6 = 0 -y = -6 y = 6 Step 3: Substitute y=6 back into the expression for x. x = 6 + 2 x = 8 The solution is x=8, y=6. iii. Given the system: x = 4 - (3)/(2)y &(1) \\ -3x + y = -1 &(2) Step 1: Substitute the expression for x from equation (1) into equation (2). -3(4 - (3)/(2)y) + y = -1 -12 + (9)/(2)y + y = -1 Step 2: Combine the y terms. -12 + (9)/(2)y + (2)/(2)y = -1 -12 + (11)/(2)y = -1 Step 3: Add 12 to both sides. (11)/(2)y = -1 + 12 (11)/(2)y = 11 Step 4: Multiply by (2)/(11) to solve for y. y = 11 × (2)/(11) y = 2 Step 5: Substitute y=2 back into equation (1) to find x. x = 4 - (3)/(2)(2) x = 4 - 3 x = 1 The solution is x=1, y=2. 2. Solve the inequalities i. (1)/(3)x - (1)/(2)(x-3) < 1 Step 1: Distribute (1)/(2). (1)/(3)x - (1)/(2)x + (3)/(2) < 1 Step 2: Find a common denominator for the x terms (LCM of 3 and 2 is 6). (2)/(6)x - (3)/(6)x + (3)/(2) < 1 -(1)/(6)x + (3)/(2) < 1 Step 3: Subtract (3)/(2) from both sides. -(1)/(6)x < 1 - (3)/(2) -(1)/(6)x < (2)/(2) - (3)/(2) -(1)/(6)x < -(1)/(2) Step 4: Multiply both sides by -6 and reverse the inequality sign. x > (-(1)/(2)) × (-6) x > 3 The solution is x > 3. ii. (x+4)/(2x+2) 1 Step 1: Subtract 1 from both sides. (x+4)/(2x+2) - 1 0 Step 2: Find a common denominator. (x+4)/(2x+2) - (2x+2)/(2x+2) 0 (x+4 - (2x+2))/(2x+2) 0 (x+4 - 2x - 2)/(2x+2) 0 (-x+2)/(2x+2) 0 Step 3: Find critical points by setting the numerator and denominator to zero. Numerator: -x+2 = 0 x = 2 Denominator: 2x+2 = 0 2x = -2 x = -1 Step 4: Test intervals (-, -1), (-1, 2], and [2, ). For x < -1 (e.g., x=-2): (-(-2)+2)/(2(-2)+2) = (4)/(-2) = -2 0. This interval is part of the solution. For -1 < x 2 (e.g., x=0): (-0+2)/(2(0)+2) = (2)/(2) = 1 0. This interval is not part of the solution. For x > 2 (e.g., x=3): (-3+2)/(2(3)+2) = (-1)/(8) 0. This interval is part of the solution. Note that x=-1 is excluded because it makes the denominator zero. x=2 is included because the inequality is . The solution is x < -1 or x 2. iii. (4)/(5)x (7)/(5) + (2)/(5)x Step 1: Subtract (2)/(5)x from both sides. (4)/(5)x - (2)/(5)x (7)/(5) (2)/(5)x (7)/(5) Step 2: Multiply both sides by (5)/(2). x (7)/(5) × (5)/(2) x (7)/(2) The solution is x (7)/(2). iv. (1)/(2)(5x - 4 x - (11)/(2)) Step 1: Multiply both sides by 2 to clear the fraction. 5x - 4 2(x - (11)/(2)) 5x - 4 2x - 11 Step 2: Subtract 2x from both sides. 5x - 2x - 4 -11 3x - 4 -11 Step 3: Add 4 to both sides. 3x -11 + 4 3x -7 Step 4: Divide by 3. x -(7)/(3) The solution is x -(7)/(3). v. 5x - 2 3x + 6 Step 1: Subtract 3x from both sides. 5x - 3x - 2 6 2x - 2 6 Step 2: Add 2 to both sides. 2x 6 + 2 2x 8 Step 3: Divide by 2. x 4 The solution is x 4. vi. x - 7 -5 Step 1: Add 7 to both sides. x -5 + 7 x 2 The solution is x 2. 3. Solve for x i. (3)/(x-2) - (4)/(x+1) = 0 Step 1: Add (4)/(x+1) to both sides. (3)/(x-2) = (4)/(x+1) Step 2: Cross-multiply. 3(x+1) = 4(x-2) 3x + 3 = 4x - 8 Step 3: Subtract 3x from both sides. 3 = 4x - 3x - 8 3 = x - 8 Step 4: Add 8 to both sides. 3 + 8 = x x = 11 Check for restrictions: x 2 and x -1. Our solution x=11 is valid. The solution is x=11. ii. (3x-5)/(9) + 2x = (1)/(3) Step 1: Find a common denominator for all terms (LCM of 9 and 3 is 9). (3x-5)/(9) + (18x)/(9) = (3)/(9) Step 2: Multiply the entire equation by 9. 3x - 5 + 18x = 3 Step 3: Combine like terms. 21x - 5 = 3 Step 4: Add 5 to both sides. 21x = 3 + 5 21x = 8 Step 5: Divide by 21. x = (8)/(21) The solution is x=(8)/(21). iii. (x)/(7) - (x)/(5) = (10)/(7) Step 1: Find a common denominator for the x terms (LCM of 7 and 5 is 35). (5x)/(35) - (7x)/(35) = (10)/(7) (5x - 7x)/(35) =