Solve these simultaneous equations:

7y+2x= 11.5

5y+3x= 9

We are given the following system of simultaneous linear equations:

1. \( 7y + 2x = 11.5 \)
2. \( 5y + 3x = 9 \)

We can solve this system using the substitution or elimination method. Here, we'll use the **elimination method**.

### Step 1: Eliminate one variable
To eliminate one variable, we can multiply each equation by appropriate constants so that the coefficients of one of the variables are the same. Let's eliminate \(x\) by multiplying the first equation by 3 and the second equation by 2:

- Multiply the first equation by 3:
  \[
  3(7y + 2x) = 3(11.5) \quad \Rightarrow \quad 21y + 6x = 34.5
  \]
- Multiply the second equation by 2:
  \[
  2(5y + 3x) = 2(9) \quad \Rightarrow \quad 10y + 6x = 18
  \]

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

\[
(21y + 6x) - (10y + 6x) = 34.5 - 18
\]

This simplifies to:

\[
11y = 16.5
\]

### Step 3: Solve for \(y\)
\[
y = \frac{16.5}{11} = 1.5
\]

### Step 4: Substitute \(y = 1.5\) into one of the original equations
We can substitute \(y = 1.5\) into the first equation:

\[
7(1.5) + 2x = 11.5
\]
\[
10.5 + 2x = 11.5
\]
\[
2x = 11.5 - 10.5
\]
\[
2x = 1
\]
\[
x = \frac{1}{2} = 0.5
\]

### Final Solution
The solution to the system of equations is:
\[
x = 0.5, \quad y = 1.5
\]

Would you like more details on any part of the process?

### Related Questions:
1. What are some other methods for solving simultaneous equations?
2. How can you solve nonlinear systems of equations?
3. Can graphing be used to solve this system of equations?
4. How does the substitution method work in solving systems of equations?
5. What is Cramer's rule, and how can it be applied to solve systems of equations?

### Tip:
When solving simultaneous equations, it's helpful to choose the method (elimination, substitution, or graphical) that seems easiest based on the structure of the equations.

Solve these simultaneous equations:
7y + 2x = 11.5
5y + 3x = 9

Learn how to solve the system of simultaneous linear equations 7y + 2x = 11.5 and 5y + 3x = 9 using the elimination method. This detailed solution walks through each step, covering key algebraic concepts. Suitable for students in Grades 8-10.

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