solve the system of equations 2x=y-5, 5x+3y=4

To solve the system of equations:

\[
1) \ 2x = y - 5
\]
\[
2) \ 5x + 3y = 4
\]

### Step 1: Solve the first equation for \( y \)
From equation (1), we can solve for \( y \) in terms of \( x \):
\[
2x = y - 5
\]
\[
y = 2x + 5
\]

### Step 2: Substitute \( y \) into the second equation
Now, substitute \( y = 2x + 5 \) into equation (2):
\[
5x + 3(2x + 5) = 4
\]
Expand the equation:
\[
5x + 6x + 15 = 4
\]
\[
11x + 15 = 4
\]
Solve for \( x \):
\[
11x = 4 - 15
\]
\[
11x = -11
\]
\[
x = -1
\]

### Step 3: Find \( y \)
Substitute \( x = -1 \) into \( y = 2x + 5 \):
\[
y = 2(-1) + 5 = -2 + 5 = 3
\]

### Solution:
The solution to the system of equations is \( x = -1 \) and \( y = 3 \).

Would you like any more details or further explanation?

Here are 5 related questions for further exploration:
1. What is the graphical interpretation of solving this system of equations?
2. How can we solve this system of equations using matrices?
3. What happens when the system of equations has no solutions?
4. Can we use substitution to solve non-linear systems?
5. How do we solve a system of three variables instead of two?

**Tip**: When solving systems of equations, substitution is often easier when one variable is already isolated.

Learn how to solve the system of equations 2x = y - 5 and 5x + 3y = 4 using the substitution method. This problem focuses on key algebraic techniques for solving linear systems of equations, making it ideal for students in Grades 8-10.

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