11x–2y=5
4x+6y=9
What value of c will make this system have the same solution?
11x–2y=5
cx=24

To find the value of \( c \) that will make the system:

1. \( 11x - 2y = 5 \)
2. \( cx = 24 \)

have the same solution as the original system:

1. \( 11x - 2y = 5 \)
2. \( 4x + 6y = 9 \),

we need to express the second equation in terms of only \( x \), so we eliminate \( y \) in both systems.

### Step 1: Solve the second equation of the original system for \( y \)

From the original system, the second equation is:
\[
4x + 6y = 9
\]

Solve for \( y \):
\[
6y = 9 - 4x
\]
\[
y = \frac{9 - 4x}{6}
\]
\[
y = \frac{3}{2} - \frac{2x}{3}
\]

### Step 2: Substitute this expression for \( y \) in the first equation of the original system

Now substitute \( y = \frac{3}{2} - \frac{2x}{3} \) into the first equation \( 11x - 2y = 5 \):

\[
11x - 2\left( \frac{3}{2} - \frac{2x}{3} \right) = 5
\]
Simplify:
\[
11x - 2 \times \frac{3}{2} + 2 \times \frac{2x}{3} = 5
\]
\[
11x - 3 + \frac{4x}{3} = 5
\]
Multiply everything by 3 to eliminate the fraction:
\[
33x - 9 + 4x = 15
\]
\[
37x = 24
\]
\[
x = \frac{24}{37}
\]

### Step 3: Substitute this value of \( x \) into the equation \( cx = 24 \)

Now, from the second system, we know that \( cx = 24 \). Substituting \( x = \frac{24}{37} \) into this equation:

\[
c \times \frac{24}{37} = 24
\]
Multiply both sides by 37:
\[
c \times 24 = 24 \times 37
\]
\[
c = 37
\]

Thus, the value of \( c \) that will make the system have the same solution is \( \boxed{37} \).

---

Would you like more details on any step? 

Here are 5 related questions for further exploration:
1. How can we verify that \( x = \frac{24}{37} \) satisfies both systems?
2. How would the solution change if the second equation had a different constant, such as \( cx = 12 \)?
3. How can you solve this system graphically?
4. What happens if the value of \( c \) is different from 37? Does the system still have a solution?
5. Can you solve this system using substitution or elimination?

**Tip:** When solving systems of equations with different forms, always try to eliminate one variable first for clarity.

Explore how to find the value of c in a system of equations: 11x - 2y = 5 and cx = 24, ensuring the same solution as 11x - 2y = 5 and 4x + 6y = 9. A step-by-step solution using substitution shows the process of eliminating variables and solving for x and c. Suitable for Grades 9-12.

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