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.