Question

1. représente la situation suivante à l’aide d’un système linéaire : le centre de recyclage remet 0,06 $ pour une petite canette métallique et 0,23 $ pour une grande canette. clotilde a rapporté 70 canettes au centre de recyclage et elle a reçu un montant de 22,35 $.

Explanation:

Step1: Define variables

Let \( x \) be the number of small cans and \( y \) be the number of large cans. We know that the total number of cans is 70, so \( x + y = 70 \). The total money received is \( 0.06x + 0.23y = 22.35 \).

Step2: Solve the system

From \( x + y = 70 \), we get \( x = 70 - y \). Substitute into the second equation: \( 0.06(70 - y) + 0.23y = 22.35 \). Expand: \( 4.2 - 0.06y + 0.23y = 22.35 \). Combine like terms: \( 4.2 + 0.17y = 22.35 \). Subtract 4.2: \( 0.17y = 18.15 \). Divide by 0.17: \( y = \frac{18.15}{0.17} = 106.76\)? Wait, no, maybe the problem is to represent the situation with a linear system. Let's re - express. Let \( x \) = small cans, \( y \) = large cans. Then:

1. \( x + y = 70 \) (total number of cans)
2. \( 0.06x + 0.23y = 22.35 \) (total money)

We can also write the first equation as \( x=70 - y \) and substitute into the second, but the task is to represent the situation. So the linear system is:
\(

$$\begin{cases}x + y=70\\0.06x + 0.23y = 22.35\end{cases}$$

\)

Answer:

The linear system representing the situation is \(\boldsymbol{

$$\begin{cases}x + y = 70\\0.06x+0.23y=22.35\end{cases}$$

}\) where \( x \) is the number of small metallic cans and \( y \) is the number of large cans.