2y=3x-2                                                                                                                                                                                                                                          6y=5x+10

You have a system of two linear equations:

1. \( 2y = 3x - 2 \)
2. \( 6y = 5x + 10 \)

Let's solve this system using substitution or elimination. We'll start by simplifying and then solving.

### Step 1: Solve for \( y \) in both equations
First, solve both equations for \( y \).

From the first equation:
\[
2y = 3x - 2
\]
Divide both sides by 2:
\[
y = \frac{3x - 2}{2}
\]

From the second equation:
\[
6y = 5x + 10
\]
Divide both sides by 6:
\[
y = \frac{5x + 10}{6}
\]

### Step 2: Set the two expressions for \( y \) equal to each other
Now, set the two expressions for \( y \) equal to each other:

\[
\frac{3x - 2}{2} = \frac{5x + 10}{6}
\]

### Step 3: Eliminate the fractions
To eliminate the fractions, multiply through by the least common denominator (6):

\[
6 \cdot \frac{3x - 2}{2} = 6 \cdot \frac{5x + 10}{6}
\]
Simplifying both sides:

\[
3(3x - 2) = 5x + 10
\]

### Step 4: Expand and solve for \( x \)
Expand the left-hand side:

\[
9x - 6 = 5x + 10
\]
Now, move all \( x \)-terms to one side and constants to the other:

\[
9x - 5x = 10 + 6
\]
Simplify:

\[
4x = 16
\]
Solve for \( x \):

\[
x = 4
\]

### Step 5: Solve for \( y \)
Now, substitute \( x = 4 \) back into one of the original expressions for \( y \). Let's use the first equation:

\[
y = \frac{3x - 2}{2}
\]
Substitute \( x = 4 \):

\[
y = \frac{3(4) - 2}{2} = \frac{12 - 2}{2} = \frac{10}{2} = 5
\]

### Solution
The solution to the system of equations is:

\[
x = 4, \quad y = 5
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How would you solve this system using the elimination method?
2. How do you graphically represent the solution to this system of equations?
3. What is the significance of the solution in terms of intersection points of the lines?
4. How would the solution change if one of the equations was multiplied by a constant?
5. Can this system of equations be represented using matrices?

**Tip:** When dealing with fractions in equations, multiplying through by the least common denominator can simplify the process significantly.

Learn how to solve a system of linear equations 2y = 3x - 2 and 6y = 5x + 10 step-by-step. Understand the methods of substitution and elimination used to find the values of x and y. Suitable for high school students and anyone interested in algebraic equations.

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