This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to your questions in English: 11. a) Simplify the following: 11. a) i) 2x - 3y + 4x + 5y Step 1: Group like terms. (2x + 4x) + (-3y + 5y) Step 2: Combine like terms. 6x + 2y The simplified expression is: 6x + 2y 11. a) ii) 3(x+2) - 2(x-1) Step 1: Distribute the numbers outside the parentheses. (3 × x) + (3 × 2) - (2 × x) - (2 × -1) 3x + 6 - 2x + 2 Step 2: Group like terms. (3x - 2x) + (6 + 2) Step 3: Combine like terms. x + 8 The simplified expression is: x + 8 11. a) iii) (1)/(2) (4x - 6y) + (1)/(3) (9x + 3y) Step 1: Distribute the fractions into the parentheses. ((1)/(2) × 4x) - ((1)/(2) × 6y) + ((1)/(3) × 9x) + ((1)/(3) × 3y) 2x - 3y + 3x + y Step 2: Group like terms. (2x + 3x) + (-3y + y) Step 3: Combine like terms. 5x - 2y The simplified expression is: 5x - 2y 11. b) Solve the following equations: 11. b) i) 3x - 5 = 10 Step 1: Add 5 to both sides of the equation. 3x - 5 + 5 = 10 + 5 3x = 15 Step 2: Divide both sides by 3. (3x)/(3) = (15)/(3) x = 5 The solution is: x = 5 11. b) ii) 2(x+3) = 14 Step 1: Divide both sides by 2. (2(x+3))/(2) = (14)/(2) x + 3 = 7 Step 2: Subtract 3 from both sides. x + 3 - 3 = 7 - 3 x = 4 The solution is: x = 4 11. b) iii) (x)/(4) + 2 = 5 Step 1: Subtract 2 from both sides of the equation. (x)/(4) + 2 - 2 = 5 - 2 (x)/(4) = 3 Step 2: Multiply both sides by 4. (x)/(4) × 4 = 3 × 4 x = 12 The solution is: x = 12 11. c) Solve the following simultaneous equations using substitution method: x + y = 7 (1) 2x - y = 2 (2) Step 1: Solve equation (1) for y. y = 7 - x (3) Step 2: Substitute equation (3) into equation (2). 2x - (7 - x) = 2 2x - 7 + x = 2 Step 3: Combine like terms and solve for x. 3x - 7 = 2 3x = 2 + 7 3x = 9 x = (9)/(3) x = 3 Step 4: Substitute the value of x back into equation (3) to find y. y = 7 - 3 y = 4 The solution is: x = 3, y = 4 11. d) Solve the following simultaneous equations using elimination method: 2x + 3y = 12 (1) x - y = 1 (2) Step 1: Multiply equation (2) by 3 to make the coefficients of y opposites. 3(x - y) = 3(1) 3x - 3y = 3 (3) Step 2: Add equation (1) and equation (3). (2x + 3y) + (3x - 3y) = 12 + 3 5x = 15 Step 3: Solve for x. x = (15)/(5) x = 3 Step 4: Substitute the value of x back into equation (2) to find y. 3 - y = 1 -y = 1 - 3 -y = -2 y = 2 The solution is: x = 3, y = 2 11. e) Solve the following quadratic equation by factorization method: x^2 + 5x + 6 = 0 Step 1: Find two numbers that multiply to 6 and add up to 5. The numbers are 2 and 3 (2 × 3 = 6 and 2 + 3 = 5). Step 2: Rewrite the middle term using these two numbers. x^2 + 2x + 3x + 6 = 0 Step 3: Factor by grouping. x(x + 2) + 3(x + 2) = 0 Step 4: Factor out the common binomial (x+2). (x + 2)(x + 3) = 0 Step 5: Set each factor equal to zero and solve for x. x + 2 = 0 or x + 3 = 0 x = -2 or x = -3 The solutions are: x = -2 or x = -3 3 done, 2 left today. You're making progress.