https://file.aiquickdraw.com/mathbot/file/1728227194741qenyi643.jpg

The image shows a system of simultaneous linear equations that needs to be solved. The system is:

\[
\begin{aligned}
3c + 4d &= 10 \\
4c + 5d &= 13
\end{aligned}
\]

We will solve this system using the method of elimination. Here's how we proceed:

### Step 1: Eliminate one variable
Multiply both equations to align the coefficients of one of the variables. Let's eliminate \( c \). We multiply the first equation by 4 and the second equation by 3 to make the coefficients of \( c \) in both equations equal:

\[
\begin{aligned}
4(3c + 4d) &= 4(10) \quad \Rightarrow \quad 12c + 16d = 40 \\
3(4c + 5d) &= 3(13) \quad \Rightarrow \quad 12c + 15d = 39
\end{aligned}
\]

### Step 2: Subtract the equations
Now, subtract the second equation from the first to eliminate \( c \):

\[
(12c + 16d) - (12c + 15d) = 40 - 39
\]

Simplifying:

\[
d = 1
\]

### Step 3: Solve for \( c \)
Substitute \( d = 1 \) into one of the original equations, say \( 3c + 4d = 10 \):

\[
3c + 4(1) = 10 \quad \Rightarrow \quad 3c + 4 = 10
\]

Solving for \( c \):

\[
3c = 6 \quad \Rightarrow \quad c = 2
\]

### Final Solution:
\[
c = 2, \quad d = 1
\]

Would you like more details or have any questions about the solution?

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Here are 5 related questions to explore further:

1. How can you solve a system of equations using substitution instead of elimination?
2. What are the conditions under which a system of linear equations has no solution?
3. How do you determine if a system of equations has infinitely many solutions?
4. Can you solve a system of three linear equations using elimination? How does the process differ?
5. What is the graphical interpretation of solving two simultaneous linear equations?

**Tip:** When using elimination, it's useful to align the coefficients of one variable by multiplying the equations before subtracting or adding them.

Solve the following simultaneous equations: 3c + 4d = 10 and 4c + 5d = 13.

Learn how to solve a system of simultaneous linear equations, including 3c + 4d = 10 and 4c + 5d = 13, using the elimination method. A detailed step-by-step solution explains how to find the values of c and d. Suitable for students in Grades 9-10.

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