`2(x+y)=3y+5`
`3x+2y=-3`
Which equivalent equation could be used to solve the system of equations above?

Options

A. `2((5+y)/(2))+2y=-3`
B. `3((5)/(2)-y)+2y=-3`
C. `3x+2(2x-5)=-3`
D. `3x+2(5-2x)=-3`

Answer

To solve the system of equations given by:

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

we will manipulate the first equation to express one variable in terms of the other and substitute it into the second equation.

### Step 1: Simplify the first equation

Start with the first equation:

\[ 2(x+y) = 3y + 5 \]

Distributing the 2 on the left side:

\[ 2x + 2y = 3y + 5 \]

### Step 2: Rearrange the first equation

Now, let's rearrange this equation to isolate \(y\):

\[ 2x + 2y - 3y = 5 \]

This simplifies to:

\[ 2x - y = 5 \]

Now, we can express \(y\) in terms of \(x\):

\[ y = 2x - 5 \]

### Step 3: Substitute \(y\) in the second equation

Now, substitute \(y = 2x - 5\) into the second equation \(3x + 2y = -3\):

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

### Step 4: Simplify the second equation

Distributing the 2:

\[ 3x + 4x - 10 = -3 \]

Combine like terms:

\[ 7x - 10 = -3 \]

### Step 5: Solve for \(x\)

Now, add 10 to both sides:

\[ 7x = 7 \]

Dividing both sides by 7 gives:

\[ x = 1 \]

### Step 6: Find \(y\)

Now substitute \(x = 1\) back into the equation \(y = 2x - 5\):

\[ y = 2(1) - 5 = 2 - 5 = -3 \]

### Final Answer

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

An equivalent equation that can be used to solve the system is:

\[ 7x - 10 = -3 \]