https://file.aiquickdraw.com/mathbot/file/1729449281367c4vqjw2r.jpg

The simultaneous equations given in the image are:

1. \( 9y + 4x = 24.5 \)
2. \( 5y + 3x = 14 \)

We can solve these equations using either substitution or elimination. Let's use the **elimination method**.

### Step 1: Make the coefficients of one variable the same
We'll eliminate \( x \). To do that, we need to multiply both equations so that the coefficient of \( x \) becomes the same in both.

- Multiply the first equation by 3:
  \[
  3(9y + 4x) = 3(24.5)
  \]
  \[
  27y + 12x = 73.5
  \]
  
- Multiply the second equation by 4:
  \[
  4(5y + 3x) = 4(14)
  \]
  \[
  20y + 12x = 56
  \]

### Step 2: Subtract the equations to eliminate \( x \)
Now, subtract the second equation from the first:
\[
(27y + 12x) - (20y + 12x) = 73.5 - 56
\]
\[
7y = 17.5
\]
\[
y = \frac{17.5}{7} = 2.5
\]

### Step 3: Substitute \( y = 2.5 \) back into one of the original equations
Substitute \( y = 2.5 \) into the second equation:
\[
5(2.5) + 3x = 14
\]
\[
12.5 + 3x = 14
\]
\[
3x = 14 - 12.5 = 1.5
\]
\[
x = \frac{1.5}{3} = 0.5
\]

### Solution:
The solution to the system of equations is:
\[
x = 0.5, \quad y = 2.5
\]

Would you like further clarification or details on any step?

Here are 5 related questions to deepen your understanding:
1. What happens if you solve the system using the substitution method instead?
2. How can you verify the solution by substituting \( x = 0.5 \) and \( y = 2.5 \) into both original equations?
3. How would this method change if the system had no solution or an infinite number of solutions?
4. Can you derive a system of equations that has only one solution, no solution, and infinitely many solutions?
5. How do you approach solving a system of non-linear simultaneous equations?

**Tip:** When solving systems of equations, always check your solution by substituting it back into the original equations to ensure it satisfies both.

Solve the simultaneous equations: 9y + 4x = 24.5 and 5y + 3x = 14.

Learn how to solve the system of linear equations 9y + 4x = 24.5 and 5y + 3x = 14 step-by-step using the elimination method. This guide walks through each step, making it easy to understand for students in Grades 8-10.

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