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.
ScanSolve AI Answer
Another one prince — let's solve it. Here are the solutions for Exercise 4 using the elimination method: 1. (a) Given the equations: 2x + y = 5 (1) x - y = 1 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (2x + y) + (x - y) = 5 + 1 3x = 6 Step 2: Solve for x. x = (6)/(3) x = 2 Step 3: Substitute x = 2 into equation (2). 2 - y = 1 y = 2 - 1 y = 1 The solution is x=2, y=1. 1. (b) Given the equations: 2x - y = 5 (1) x + y = 4 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (2x - y) + (x + y) = 5 + 4 3x = 9 Step 2: Solve for x. x = (9)/(3) x = 3 Step 3: Substitute x = 3 into equation (2). 3 + y = 4 y = 4 - 3 y = 1 The solution is x=3, y=1. 2. (a) Given the equations: x + 2y = 11 (1) 2x - 2y = 10 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (x + 2y) + (2x - 2y) = 11 + 10 3x = 21 Step 2: Solve for x. x = (21)/(3) x = 7 Step 3: Substitute x = 7 into equation (1). 7 + 2y = 11 2y = 11 - 7 2y = 4 Step 4: Solve for y. y = (4)/(2) y = 2 The solution is x=7, y=2. 2. (b) Given the equations: 2a - b = 1 (1) 3a + 2b = 19 (2) Step 1: Multiply equation (1) by 2 to make the coefficient of b equal to -2. 2(2a - b) = 2(1) 4a - 2b = 2 (3) Step 2: Add equation (3) and equation (2) to eliminate b. (4a - 2b) + (3a + 2b) = 2 + 19 7a = 21 Step 3: Solve for a. a = (21)/(7) a = 3 Step 4: Substitute a = 3 into equation (1). 2(3) - b = 1 6 - b = 1 b = 6 - 1 b = 5 The solution is a=3, b=5. 3. (a) Given the equations: 5p + 2q = 18 (1) p + 2q = 2 (2) Step 1: Subtract equation (2) from equation (1) to eliminate q. (5p + 2q) - (p + 2q) = 18 - 2 4p = 16 Step 2: Solve for p. p = (16)/(4) p = 4 Step 3: Substitute p = 4 into equation (2). 4 + 2q = 2 2q = 2 - 4 2q = -2 Step 4: Solve for q. q = (-2)/(2) q = -1 The solution is p=4, q=-1. 3. (b) Given the equations: 3p + 5k = 30 (1) 2p + 3k = 19 (2) Step 1: Multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of p equal. 2(3p + 5k) = 2(30) 6p + 10k = 60 (3) 3(2p + 3k) = 3(19) 6p + 9k = 57 (4) Step 2: Subtract equation (4) from equation (3) to eliminate p. (6p + 10k) - (6p + 9k) = 60 - 57 k = 3 Step 3: Substitute k = 3 into equation (1). 3p + 5(3) = 30 3p + 15 = 30 3p = 30 - 15 3p = 15 Step 4: Solve for p. p = (15)/(3) p = 5 The solution is p=5, k=3. 4. (a) Given the equations: 4m + 3n = 28 (1) 6m + 2n = 32 (2) Step 1: Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of m equal. 3(4m + 3n) = 3(28) 12m + 9n = 84 (3) 2(6m + 2n) = 2(32) 12m + 4n = 64 (4) Step 2: Subtract equation (4) from equation (3) to eliminate m. (12m + 9n) - (12m + 4n) = 84 - 64 5n = 20 Step 3: Solve for n. n = (20)/(5) n = 4 Step 4: Substitute n = 4 into equation (1). 4m + 3(4) = 28 4m + 12 = 28 4m = 28 - 12 4m = 16 Step 5: Solve for m. m = (16)/(4) m = 4 The solution is m=4, n=4. 4. (b) Given the equations: 3x + 5y = 7 (1) 4x + 3y = 2 (2) Step 1: Multiply equation (1) by 4 and equation (2) by 3 to make the coefficients of x equal. 4(3x + 5y) = 4(7) 12x + 20y = 28 (3) 3(4x + 3y) = 3(2) 12x + 9y = 6 (4) Step 2: Subtract equation (4) from equation (3) to eliminate x. (12x + 20y) - (12x + 9y) = 28 - 6 11y = 22 Step 3: Solve for y. y = (22)/(11) y = 2 Step 4: Substitute y = 2 into equation (1). 3x + 5(2) = 7 3x + 10 = 7 3x = 7 - 10 3x = -3 Step 5: Solve for x. x = (-3)/(3) x = -1 The solution is x=-1, y=2. Send me the next one 📸