https://file.aiquickdraw.com/mathbot/file/17288365217484gbb4ida.jpg

The system of simultaneous equations provided in the image is:

\[
7x + 5y = 8
\]
\[
3x - 2y = -9
\]

We can solve this system algebraically using either substitution or elimination. I'll use the elimination method here. The goal is to eliminate one variable and solve for the other.

### Step 1: Multiply both equations to make the coefficients of one variable (e.g., \(y\)) the same
To eliminate \(y\), we need the coefficients of \(y\) in both equations to be equal (in absolute value). We can do this by multiplying the first equation by 2 and the second equation by 5.

\[
2(7x + 5y) = 2(8) \quad \text{(Multiply the first equation by 2)}
\]
\[
14x + 10y = 16
\]
\[
5(3x - 2y) = 5(-9) \quad \text{(Multiply the second equation by 5)}
\]
\[
15x - 10y = -45
\]

### Step 2: Add the two equations to eliminate \(y\)
Now, we add the two equations:

\[
(14x + 10y) + (15x - 10y) = 16 + (-45)
\]
\[
14x + 15x = -29
\]
\[
29x = -29
\]

### Step 3: Solve for \(x\)
\[
x = \frac{-29}{29} = -1
\]

### Step 4: Substitute \(x = -1\) into one of the original equations to find \(y\)
Substitute \(x = -1\) into the first equation:

\[
7(-1) + 5y = 8
\]
\[
-7 + 5y = 8
\]
\[
5y = 8 + 7
\]
\[
5y = 15
\]
\[
y = \frac{15}{5} = 3
\]

### Solution:
The solution to the system of equations is:
\[
x = -1, \quad y = 3
\]

Would you like more details or have any further questions?

Here are 5 related questions you might explore:
1. How would the substitution method work for this system of equations?
2. Can this system of equations be solved graphically? How?
3. What happens if the system has no solution or infinitely many solutions?
4. How can we verify the solution by plugging the values of \(x\) and \(y\) back into the equations?
5. What are some real-world applications of solving simultaneous equations?

**Tip:** When using elimination, make sure the variables you're eliminating have coefficients that are opposites to avoid mistakes when adding or subtracting equations!

Solve the following simultaneous equations algebraically:
7x + 5y = 8
3x - 2y = -9

Learn how to solve the simultaneous equations 7x + 5y = 8 and 3x - 2y = -9 using the elimination method. This solution is suitable for students in Grades 9-11, covering key concepts such as algebraic manipulation and solving linear equations.

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