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
Here are the solutions to the problems: 1. Simplify (3a+4b)/(10) + (5a-b)/(4) Step 1: Find the least common multiple (LCM) of the denominators 10 and 4. The LCM is 20. Step 2: Rewrite each fraction with the common denominator. (3a+4b)/(10) = (2(3a+4b))/(2 × 10) = (6a+8b)/(20) (5a-b)/(4) = (5(5a-b))/(5 × 4) = (25a-5b)/(20) Step 3: Add the fractions. (6a+8b)/(20) + (25a-5b)/(20) = (6a+8b+25a-5b)/(20) Step 4: Combine like terms in the numerator. ((6a+25a) + (8b-5b))/(20) = (31a+3b)/(20) The simplified expression is (31a+3b)/(20). 2. Simplify (6y-5x)/(10) - (2x+y)/(5) + (x)/(3) Step 1: Find the LCM of the denominators 10, 5, and 3. The LCM is 30. Step 2: Rewrite each fraction with the common denominator. (6y-5x)/(10) = (3(6y-5x))/(3 × 10) = (18y-15x)/(30) (2x+y)/(5) = (6(2x+y))/(6 × 5) = (12x+6y)/(30) (x)/(3) = (10x)/(10 × 3) = (10x)/(30) Step 3: Combine the fractions. (18y-15x)/(30) - (12x+6y)/(30) + (10x)/(30) = ((18y-15x) - (12x+6y) + 10x)/(30) Step 4: Distribute the negative sign and combine like terms in the numerator. (18y-15x-12x-6y+10x)/(30) = ((18y-6y) + (-15x-12x+10x))/(30) (12y - 17x)/(30) The simplified expression is (12y-17x)/(30). 3. Simplify (5t)/(9) + (t-3)/(4) - (t-3)/(2) Step 1: Find the LCM of the denominators 9, 4, and 2. The LCM is 36. Step 2: Rewrite each fraction with the common denominator. (5t)/(9) = (4(5t))/(4 × 9) = (20t)/(36) (t-3)/(4) = (9(t-3))/(9 × 4) = (9t-27)/(36) (t-3)/(2) = (18(t-3))/(18 × 2) = (18t-54)/(36) Step 3: Combine the fractions. (20t)/(36) + (9t-27)/(36) - (18t-54)/(36) = (20t + (9t-27) - (18t-54))/(36) Step 4: Distribute the negative sign and combine like terms in the numerator. (20t+9t-27-18t+54)/(36) = ((20t+9t-18t) + (-27+54))/(36) (11t+27)/(36) The simplified expression is (11t+27)/(36). 4. Simplify (u+5)/(4) - (3u-5)/(2) Step 1: Find the LCM of the denominators 4 and 2. The LCM is 4. Step 2: Rewrite the second fraction with the common denominator. (3u-5)/(2) = (2(3u-5))/(2 × 2) = (6u-10)/(4) Step 3: Subtract the fractions. (u+5)/(4) - (6u-10)/(4) = ((u+5) - (6u-10))/(4) Step 4: Distribute the negative sign and combine like terms in the numerator. (u+5-6u+10)/(4) = ((u-6u) + (5+10))/(4) (-5u+15)/(4) The simplified expression is (-5u+15)/(4). 5. Simplify (x-3)/(3) - (x+2)/(5) Step 1: Find the LCM of the denominators 3 and 5. The LCM is 15. Step 2: Rewrite each fraction with the common denominator. (x-3)/(3) = (5(x-3))/(5 × 3) = (5x-15)/(15) (x+2)/(5) = (3(x+2))/(3 × 5) = (3x+6)/(15) Step 3: Subtract the fractions. (5x-15)/(15) - (3x+6)/(15) = ((5x-15) - (3x+6))/(15) Step 4: Distribute the negative sign and combine like terms in the numerator. (5x-15-3x-6)/(15) = ((5x-3x) + (-15-6))/(15) (2x-21)/(15) The simplified expression is (2x-21)/(15). That's 2 down. 3 left today — send the next one.