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 the simultaneous equations and word problems: 1. We have the equations: x+y=-1 (1) 2x-y=-3 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (x+y) + (2x-y) = -1 + (-3) 3x = -4 x = -(4)/(3) Step 2: Substitute the value of x into equation (1). -(4)/(3) + y = -1 y = -1 + (4)/(3) y = -(3)/(3) + (4)/(3) y = (1)/(3) The solution is: x = -(4)/(3), y = (1)/(3) 2. We have the equations: 3x+y=7 (1) 6x-y=25 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (3x+y) + (6x-y) = 7 + 25 9x = 32 x = (32)/(9) Step 2: Substitute the value of x into equation (1). 3((32)/(9)) + y = 7 (32)/(3) + y = 7 y = 7 - (32)/(3) y = (21)/(3) - (32)/(3) y = -(11)/(3) The solution is: x = (32)/(9), y = -(11)/(3) 3. We have the equations: 3x+2y=8 (1) 2x-y=3 (2) Step 1: Multiply equation (2) by 2 to make the coefficients of y opposites. 2(2x-y) = 2(3) 4x-2y=6 (3) Step 2: Add equation (1) and equation (3) to eliminate y. (3x+2y) + (4x-2y) = 8 + 6 7x = 14 x = 2 Step 3: Substitute the value of x into equation (2). 2(2) - y = 3 4 - y = 3 -y = 3 - 4 -y = -1 y = 1 The solution is: x = 2, y = 1 4. We have the equations: 3x+2y=7 (1) 4x+3y=15 (2) Step 1: Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of y equal. 3(3x+2y) = 3(7) 9x+6y=21 (3) 2(4x+3y) = 2(15) 8x+6y=30 (4) Step 2: Subtract equation (4) from equation (3) to eliminate y. (9x+6y) - (8x+6y) = 21 - 30 x = -9 Step 3: Substitute the value of x into equation (1). 3(-9) + 2y = 7 -27 + 2y = 7 2y = 7 + 27 2y = 34 y = 17 The solution is: x = -9, y = 17 5. Let the first number be x and the second number be y. From the problem statement, we can form two equations: x+y=72 (1) y=3x (2) Step 1: Substitute equation (2) into equation (1). x + (3x) = 72 4x = 72 x = (72)/(4) x = 18 Step 2: Substitute the value of x back into equation (2). y = 3(18) y = 54 The two numbers are: 18 and 54 6. Let the number of boys be B and the number of girls be G. From the problem statement, we can form two equations: B+G=32 (1) B=G+8 (2) Step 1: Substitute equation (2) into equation (1). (G+8) + G = 32 2G + 8 = 32 2G = 32 - 8 2G = 24 G = (24)/(2) G = 12 Step 2: Substitute the value of G back into equation (2). B = 12 + 8 B = 20 The number of boys in the class is: 20 boys 3 done, 2 left today. You're making progress.