1. **Stating the problem:** Given the expressions: $$x=5a+7b+9c,\quad y=b-3a-4c,\quad z=c-2b+a,$$ we need to solve the following: a) Subtract $a-c$ from $z$. b) Show that $x+y+z=3(a+2b+2c)$. c) Prove that $(x + y)-(a + 10b+4c)$ equals a certain expression (we will simplify it). --- 2. **Step a: Subtract $a-c$ from $z$** We calculate: $$z - (a - c) = (c - 2b + a) - (a - c) = c - 2b + a - a + c = 2c - 2b.$$ So, the result is: $$2c - 2b.$$ --- 3. **Step b: Show that $x + y + z = 3(a + 2b + 2c)$** Add $x$, $y$, and $z$: $$x + y + z = (5a + 7b + 9c) + (b - 3a - 4c) + (c - 2b + a).$$ Combine like terms: - For $a$: $5a - 3a + a = 3a$ - For $b$: $7b + b - 2b = 6b$ - For $c$: $9c - 4c + c = 6c$ So, $$x + y + z = 3a + 6b + 6c = 3(a + 2b + 2c).$$ This proves the statement. --- 4. **Step c: Prove that $(x + y) - (a + 10b + 4c)$ equals a simplified expression** First, find $x + y$: $$x + y = (5a + 7b + 9c) + (b - 3a - 4c) = (5a - 3a) + (7b + b) + (9c - 4c) = 2a + 8b + 5c.$$ Now subtract $(a + 10b + 4c)$: $$(x + y) - (a + 10b + 4c) = (2a + 8b + 5c) - (a + 10b + 4c) = (2a - a) + (8b - 10b) + (5c - 4c) = a - 2b + c.$$ So, $$(x + y) - (a + 10b + 4c) = a - 2b + c.$$