Question $$ 2 / $$ Vraag 2 The supply function for a commodity takes the form $$ q^{3}(p)=a p^{2}+b p+c $$, 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 aanbodfuniksie vir ' $$ n $$ produk het die vorm $$ q^{3}(p)=a p^{2}+b p+c $$, vir die konstantes

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To determine the constants $$ a, b, c $$ in the supply function $$ q^{3}(p) = a p^{2} + b p + c $$ using matrix reduction (Gaussian elimination), we can set up a system of equations based on the given information.

Given:
1. When $$ p = 1 $$, the quantity supplied is 5.
2. When $$ p = 2 $$, the quantity supplied is 12.
3. When $$ p = 3 $$, the quantity supplied is 23.

We can set up the following system of equations:
1. $$ a + b + c = 5 $$
2. $$ 4a + 2b + c = 12 $$
3. $$ 9a + 3b + c = 23 $$

Let's represent this system of equations in matrix form and then use Gaussian elimination to solve for $$ a, b, c $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}1	imes a+1	imes b+1	imes c=5\4a+2b+1	imes c=12\9a+3b+1	imes c=23\end{array}ight.$$
- step1: Calculate:
  $$\left\{ \begin{array}{l}a+b+c=5\4a+2b+c=12\9a+3b+c=23\end{array}ight.$$
- step2: Solve the equation:
  $$\left\{ \begin{array}{l}a=5-b-c\4a+2b+c=12\9a+3b+c=23\end{array}ight.$$
- step3: Substitute the value of $$a:$$
  $$\left\{ \begin{array}{l}4\left(5-b-cight)+2b+c=12\9\left(5-b-cight)+3b+c=23\end{array}ight.$$
- step4: Simplify:
  $$\left\{ \begin{array}{l}20-2b-3c=12\45-6b-8c=23\end{array}ight.$$
- step5: Solve the equation:
  $$\left\{ \begin{array}{l}b=\frac{8-3c}{2}\45-6b-8c=23\end{array}ight.$$
- step6: Substitute the value of $$b:$$
  $$45-6	imes \frac{8-3c}{2}-8c=23$$
- step7: Simplify:
  $$21+c=23$$
- step8: Move the constant to the right side:
  $$c=23-21$$
- step9: Subtract the numbers:
  $$c=2$$
- step10: Substitute the value of $$c:$$
  $$b=\frac{8-3	imes 2}{2}$$
- step11: Calculate:
  $$b=1$$
- step12: Substitute the values of $$b,c:$$
  $$a=5-1-2$$
- step13: Calculate:
  $$a=2$$
- step14: Calculate:
  $$\left\{ \begin{array}{l}a=2\b=1\c=2\end{array}ight.$$
- step15: Check the solution:
  $$\left\{ \begin{array}{l}a=2\b=1\c=2\end{array}ight.$$
- step16: Rewrite:
  $$\left(a,b,cight) = \left(2,1,2ight)$$
The solution to the system of equations is $$ a = 2, b = 1, c = 2 $$.

Therefore, the supply function is $$ q^{3}(p) = 2p^{2} + p + 2 $$.
The constants are $$ a = 2 $$, $$ b = 1 $$, and $$ c = 2 $$. The supply function is $$ q^{3}(p) = 2p^{2} + p + 2 $$.

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