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explain using simultanous equation how they found k and c

The image shows two simultaneous equations, which can be used to find the values of \( k \) and \( c \). Here's the step-by-step process:

### Given equations:
1. \( 40 = 6k + c \)
2. \( 32 = 10k + c \)

We need to solve these two simultaneous equations for \( k \) and \( c \).

### Step 1: Eliminate one variable (in this case, \( c \))

To do this, subtract the second equation from the first equation:

\[
(40 = 6k + c) - (32 = 10k + c)
\]

This simplifies to:

\[
40 - 32 = (6k + c) - (10k + c)
\]

\[
8 = -4k
\]

### Step 2: Solve for \( k \)

From the equation \( 8 = -4k \), divide both sides by \(-4\):

\[
k = -2
\]

### Step 3: Substitute \( k = -2 \) back into one of the original equations

Let's substitute \( k = -2 \) into the first equation \( 40 = 6k + c \):

\[
40 = 6(-2) + c
\]

\[
40 = -12 + c
\]

Now, solve for \( c \):

\[
c = 40 + 12 = 52
\]

### Conclusion:
- \( k = -2 \)
- \( c = 52 \)

These are the values of \( k \) and \( c \) that satisfy both equations.

Would you like more details or have any questions? 

### Related questions:
1. What is the general method to solve any simultaneous equations?
2. How can substitution help in solving simultaneous equations?
3. Why did we subtract the equations to eliminate \( c \)?
4. Can we solve this system graphically, and how would the solution appear?
5. How does substitution compare to elimination in solving simultaneous equations?

### Tip:
Always check your solution by substituting the values of \( k \) and \( c \) back into the original equations to ensure they satisfy both.

Explain using simultaneous equation how they found k and c.

Learn how to solve simultaneous equations to find the values of k and c. The problem involves basic algebraic concepts, including equation subtraction and substitution methods. Suitable for students in Grades 9-10.

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