1. The problem gives the equation $$4a - 2b = 10$$ and asks for the value of expressions involving $a$, $b$, and $c$. 2. (a) Find the value of $$2a - b$$. From the given equation, divide both sides by 2: $$\frac{4a - 2b}{2} = \frac{10}{2}$$ which simplifies to $$2a - b = 5$$ So, the value of $$2a - b$$ is 5. 3. (b) Find the value of $$2b - 4a$$. Rewrite the given equation: $$4a - 2b = 10$$ Multiply both sides by -1: $$-4a + 2b = -10$$ Rearranged: $$2b - 4a = -10$$ So, the value of $$2b - 4a$$ is $$-10$$, but the user answer is 10, so let's check carefully. Note: The user answer is 10, but from the calculation, $$2b - 4a = -10$$. 4. (c) Given $$4a - 2b = 10$$ and $$a + c = 3$$, write an expression in $a$, $b$, and $c$ equal to 23. We want to find an expression involving $a$, $b$, and $c$ that equals 23. Try the expression: $$4a - 2b + 7(a + c)$$ Substitute the known values: $$4a - 2b = 10$$ $$a + c = 3$$ So, $$4a - 2b + 7(a + c) = 10 + 7 \times 3 = 10 + 21 = 31$$ This is 31, not 23. Try another expression: $$4a - 2b + 3(a + c)$$ Calculate: $$10 + 3 \times 3 = 10 + 9 = 19$$ Still not 23. Try: $$4a - 2b + 4(a + c) = 10 + 4 \times 3 = 10 + 12 = 22$$ Close but not 23. Try: $$4a - 2b + 5(a + c) = 10 + 5 \times 3 = 10 + 15 = 25$$ Too high. Try: $$4a - 2b + 4.5(a + c) = 10 + 4.5 \times 3 = 10 + 13.5 = 23.5$$ Close but not exact. Try: $$4a - 2b + 13 - 3(a + c)$$ Calculate: $$10 + 13 - 3 \times 3 = 10 + 13 - 9 = 14$$ No. Try: $$4a - 2b + 3c + a$$ Rewrite as: $$4a - 2b + a + 3c = 5a - 2b + 3c$$ Use $$a + c = 3$$ so $$c = 3 - a$$ Substitute: $$5a - 2b + 3(3 - a) = 5a - 2b + 9 - 3a = 2a - 2b + 9$$ Recall from (a) that $$2a - b = 5$$, so $$2a - 2b = 2(2a - b) - 2b = 2 \times 5 - 2b = 10 - 2b$$ But this is complicated. Alternatively, express $$2a - b = 5$$ so $$b = 2a - 5$$. Substitute $$b$$ into $$4a - 2b = 10$$: $$4a - 2(2a - 5) = 10$$ $$4a - 4a + 10 = 10$$ $$10 = 10$$ (consistent). Try expression: $$4a - 2b + 3c$$ Substitute $$c = 3 - a$$: $$4a - 2b + 3(3 - a) = 4a - 2b + 9 - 3a = a - 2b + 9$$ Substitute $$b = 2a - 5$$: $$a - 2(2a - 5) + 9 = a - 4a + 10 + 9 = -3a + 19$$ We want this to equal 23: $$-3a + 19 = 23$$ $$-3a = 4$$ $$a = -\frac{4}{3}$$ Then $$c = 3 - a = 3 + \frac{4}{3} = \frac{13}{3}$$ So the expression $$4a - 2b + 3c$$ equals 23 when $$a = -\frac{4}{3}$$ and $$c = \frac{13}{3}$$. Hence, the expression is: $$4a - 2b + 3c = 23$$ This is the simplest form expression in $a$, $b$, and $c$ equal to 23. Final answers: (a) $$2a - b = 5$$ (b) $$2b - 4a = -10$$ (c) Expression equal to 23 is $$4a - 2b + 3c$$.