Question

The lemonade sales at a baseball game are described as follows:

The number of small lemonades purchased is the number of mediums sold plus double the number of larges sold.
The total number of all sizes sold is 70.
One-and-a-half times the number of smalls purchased plus twice the number of mediums sold is 100.
Use a system of equations and its matrix representation to determine the number of small, medium, and large
lemonades sold

Answer 1

`Large => L=10`

`Medium => M=20`

`small => S=40`

Explanation:

The number of small lemonades purchased is the number of mediums sold plus double the number of larges sold:

`S=M+2L => S-M-2L = 0`

The total number of all sizes sold is 70:

`S+M+L = 70`

One-and-a-half times the number of smalls purchased plus twice the number of mediums sold is 100:

`1,5S+2M = 100`

The system of equations is:

`S-M-2L = 0\\S+M+L = 70\\1,5S+2M = 100`

Matrix to solve by gauss-jordan elimination:

`\left[\begin{array}{cccc}1&-1&-2&0\\1&1&1&70\\1.5&2&0&100\end{array}\right]`

Solving from first row:

`\left[\begin{array}{cccc}1&-1&-2&0\\0&2&3&70\\0&3.5&3&100\end{array}\right]`

Solving from second row:

`\left[\begin{array}{cccc}1&-1&-2&0\\0&2&3&70\\0&0&-4.5&-45\end{array}\right]`

From this:

`-4.5 L=-45 => L=10`

`2 M+3*10=70 => M=20`

`S -20-2*10=0 => S=40`