12000=120*Cc*​+192*Cw*​15000=96��+160×��×(1+�×(4−2.4))15000=96*Cc*​+160×*Cw*​×(1+*k*×(4−2.4))

Assuming �=0.2*k*=0.2 (a reasonable assumption), substitute into the equations and solve for ��*Cc*​ and ��*Cw*​.


To solve for \( C_c \) and \( C_w \), we will substitute \( k = 0.2 \) into the given equations and solve the system of equations.

The two given equations are:

1. \[ 12000 = 120C_c + 192C_w \]
2. \[ 15000 = 96C_c + 160C_w \times (1 + k \times (4 - 2.4)) \]

Substitute \( k = 0.2 \) into the second equation:

\[
15000 = 96C_c + 160C_w \times (1 + 0.2 \times (4 - 2.4))
\]

Now simplify the expression inside the parentheses:

\[
15000 = 96C_c + 160C_w \times (1 + 0.2 \times 1.6)
\]

\[
15000 = 96C_c + 160C_w \times (1 + 0.32)
\]

\[
15000 = 96C_c + 160C_w \times 1.32
\]

\[
15000 = 96C_c + 211.2C_w
\]

Now we have a system of linear equations:

1. \[ 12000 = 120C_c + 192C_w \]
2. \[ 15000 = 96C_c + 211.2C_w \]

Let's solve this system for \( C_c \) and \( C_w \). We can use the substitution or elimination method. I'll use the elimination method to solve for \( C_c \) and \( C_w \).The solutions for the variables \( C_c \) and \( C_w \) are:

- \( C_c = -50 \)
- \( C_w = 93.75 \)

This means that:

- \( C_c \) (the coefficient for \( C_c \)) is \(-50\).
- \( C_w \) (the coefficient for \( C_w \)) is \(93.75\).

Would you like more details on how this solution was derived, or do you have any questions?

### Related Questions:
1. How does changing the value of \( k \) affect the solution for \( C_c \) and \( C_w \)?
2. What is the significance of the negative value for \( C_c \) in this context?
3. How would you solve the system using the substitution method?
4. How can this system of equations be visualized graphically?
5. What other methods are there to solve linear equations, and when are they preferred?

**Tip:** When solving systems of equations, always check if the equations are independent to ensure a unique solution.

Learn how to solve a system of linear equations using the substitution method. This problem involves two equations with two variables, where we substitute a given value of k and solve for C_c and C_w. Suitable for high school students studying algebra and linear equations.

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