Question $$ 2 / $$ Vraag 2 The supply function for a commodity takes the form $$ q^{s}(p)=a p^{2}+b p+c $$, for some constants $$ a, b, c $$. When $$ p=1 $$, the quantity supplied is 5 ; when $$ p=2 $$, the quantity supplied is 12 ; when $$ p=3 $$, the quantity supplied is 23 . Use matrix reduction (Gaussian elimination) to determine the constants $$ a, b, c $$. Give the supply funcrtion. / Die aanbodfunksie vir 'n produk het die vorm $$ q^{s}(p)=a p^{2}+b p+c $$, vir die konstantes

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We are given that the supply function is
$$
q^s(p)=a p^2+b p+c,
$$
with three conditions:
$$
\begin{cases}
a(1)^2+b(1)+c=5,\[1mm]
a(2)^2+b(2)+c=12,\[1mm]
a(3)^2+b(3)+c=23.
\end{cases}
$$
This yields the system:
$$
\begin{aligned}
a+b+c&=5,\[1mm]
4a+2b+c&=12,\[1mm]
9a+3b+c&=23.
\end{aligned}
$$

**Step 1.** Subtract the first equation from the second:
$$
(4a+2b+c)-(a+b+c)=12-5,
$$
which simplifies to:
$$
3a+b=7. \tag{1}
$$

**Step 2.** Subtract the first equation from the third:
$$
(9a+3b+c)-(a+b+c)=23-5,
$$
which simplifies to:
$$
8a+2b=18.
$$
Divide through by 2:
$$
4a+b=9. \tag{2}
$$

**Step 3.** Subtract equation (1) from equation (2):
$$
(4a+b)-(3a+b)=9-7,
$$
which gives:
$$
a=2.
$$

**Step 4.** Substitute $$a=2$$ into equation (1):
$$
3(2)+b=7 \quad \Longrightarrow \quad 6+b=7,
$$
so:
$$
b=1.
$$

**Step 5.** Substitute $$a=2$$ and $$b=1$$ into the first equation:
$$
2+1+c=5 \quad \Longrightarrow \quad c=2.
$$

Thus, the constants are $$a=2$$, $$b=1$$, and $$c=2$$.

The supply function is
$$
q^s(p)=2p^2+p+2.
$$
The constants are $$ a = 2 $$, $$ b = 1 $$, and $$ c = 2 $$. The supply function is $$ q^s(p) = 2p^2 + p + 2 $$.

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