Some simplifications of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form. Observe each of them and see if there is a mistake. If you think there is a mistake, try to explain what might have gone wrong. Then, simplify it correctly.
3a + 2b = 5
False. 3a + 2b cannot be simplified further as a and b are unlike terms.
3b - 2b - b = 0
True. 3b - 2b - b = (3 - 2 - 1)b = 0.
6(p + 2) = 6p + 8
False. 6(p + 2) = 6p + 12.
(4x + 3y) - (3x + 4y) = x + y
False. (4x + 3y) - (3x + 4y) = 4x + 3y - 3x - 4y = x - y.
5 - (2 - 6z) = 3 - 6z
False. 5 - (2 - 6z) = 5 - 2 + 6z = 3 + 6z.
2 + (x + 3) = 2x - 6
False. 2 + (x + 3) = 2 + x + 3 = x + 5.
2y + (3y - 6) = -y + 6
False. 2y + (3y - 6) = 5y - 6.
7p - p + 5q - 2q = 7p + 3q
False. 7p - p + 5q - 2q = 6p + 3q.
5(2w + 3x + 4w) = 10w + 15x + 20w
False. 5(2w + 3x + 4w) = 5(6w + 3x) = 30w + 15x.
3j + 6k + 9h + 12 = 3(j + 2k + 3h + 4)
True. This is correct factorization.
4(2r + 3s + 5) = -20 - 8r - 12s
False. 4(2r + 3s + 5) = 8r + 12s + 20.
Explanation
(1) Unlike terms cannot be combined. The expression is already in simplest form. (2) All terms are like terms. 3b - 2b - b = 0b = 0. (3) Distribute 6 to both terms: 6 × p + = 6p + 12. (4) When subtracting brackets, distribute the negative sign to all terms inside. (5) Subtracting a bracket means distributing the negative: -(2 - 6z) = -2 + 6z. (6) Simply remove the brackets and add: 2 + x + 3 = x + 5. (7) Combine like terms: 2y + 3y = 5y, and -6 remains. (8) Combine like terms: (7p - p) + (5q - 2q) = 6p + 3q. (9) First combine like terms inside: 2w + 4w = 6w. Then distribute: w + x = 30w + 15x. (10) Each term has a factor of 3: 3j = 3(j), 6k = 3(2k), 9h = 3(3h), 12 = 3(4). (11) Distribute 4 to all terms: r + s + = 8r + 12s + 20.