solve 5x = 6y - 4, 2y = 3x + 4 following the systems of equations

To solve the system of equations:

\[
1) \ 5x = 6y - 4
\]
\[
2) \ 2y = 3x + 4
\]

We can use substitution or elimination. Here, I'll use substitution. Let's solve the second equation for \( y \) and substitute it into the first equation.

### Step 1: Solve the second equation for \( y \)
From the second equation:
\[
2y = 3x + 4
\]
Divide both sides by 2:
\[
y = \frac{3x + 4}{2}
\]

### Step 2: Substitute \( y = \frac{3x + 4}{2} \) into the first equation

Substitute this expression for \( y \) into the first equation:
\[
5x = 6y - 4
\]
\[
5x = 6\left(\frac{3x + 4}{2}\right) - 4
\]
Simplify the right-hand side:
\[
5x = 3(3x + 4) - 4
\]
\[
5x = 9x + 12 - 4
\]
\[
5x = 9x + 8
\]

### Step 3: Solve for \( x \)

Now, subtract \( 9x \) from both sides:
\[
5x - 9x = 8
\]
\[
-4x = 8
\]
Divide both sides by -4:
\[
x = -2
\]

### Step 4: Substitute \( x = -2 \) into the expression for \( y \)

Substitute \( x = -2 \) into the expression for \( y \):
\[
y = \frac{3(-2) + 4}{2}
\]
\[
y = \frac{-6 + 4}{2}
\]
\[
y = \frac{-2}{2}
\]
\[
y = -1
\]

### Solution:

The solution to the system of equations is:
\[
x = -2, \ y = -1
\]

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Do you want more details on any step or have any questions? Here are some follow-up questions:

1. What is another method to solve a system of equations?
2. How can you check if the solution to a system of equations is correct?
3. What is the elimination method for solving systems of equations?
4. How would graphing help solve a system of linear equations?
5. Can systems of equations have no solution or infinite solutions?

**Tip:** When solving systems of equations, double-check your solution by substituting the values back into both original equations to ensure they hold true.

Learn how to solve the system of equations 5x = 6y - 4 and 2y = 3x + 4 using the substitution method. This step-by-step solution walks through algebraic techniques suitable for students in Grades 9-10.

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