`2x+y=7-2y`
`5y-x=5-4x`
If `(x, y)` is a solution to the above system of equations, what is the vlaue of `(y+1)/(x)`?

Options

A. `-11`
B. `-(1)/(2)`
C. 2
D. 20

Answer

To solve the system of equations given by:

1. \( 2x + y = 7 - 2y \)
2. \( 5y - x = 5 - 4x \)

we will follow these steps:

### Step 1: Rearranging the first equation
Start with the first equation:
\[ 2x + y = 7 - 2y \]

Add \( 2y \) to both sides:
\[ 2x + y + 2y = 7 \]
This simplifies to:
\[ 2x + 3y = 7 \]

### Step 2: Rearranging the second equation
Now, take the second equation:
\[ 5y - x = 5 - 4x \]

Add \( 4x \) to both sides:
\[ 5y - x + 4x = 5 \]
This simplifies to:
\[ 3x + 5y = 5 \]

### Step 3: Write the system of equations
Now we have the system of equations:
1. \( 2x + 3y = 7 \) (Equation 1)
2. \( 3x + 5y = 5 \) (Equation 2)

### Step 4: Eliminate one variable
To eliminate one variable, we can multiply the first equation by 3 and the second equation by 2 to make the coefficients of \( x \) the same:

Multiplying Equation 1 by 3:
\[ 3(2x + 3y) = 3(7) \]
This gives:
\[ 6x + 9y = 21 \] (Equation 3)

Multiplying Equation 2 by 2:
\[ 2(3x + 5y) = 2(5) \]
This gives:
\[ 6x + 10y = 10 \] (Equation 4)

### Step 5: Subtract the equations
Now subtract Equation 3 from Equation 4:
\[ (6x + 10y) - (6x + 9y) = 10 - 21 \]
This simplifies to:
\[ 10y - 9y = -11 \]
Thus:
\[ y = -11 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now substitute \( y = -11 \) back into Equation 1:
\[ 2x + 3(-11) = 7 \]
This simplifies to:
\[ 2x - 33 = 7 \]
Adding 33 to both sides:
\[ 2x = 40 \]
Dividing by 2:
\[ x = 20 \]

### Step 7: Calculate \( \frac{y + 1}{x} \)
Now we need to find the value of \( \frac{y + 1}{x} \):
\[ \frac{y + 1}{x} = \frac{-11 + 1}{20} = \frac{-10}{20} = -\frac{1}{2} \]

Thus, the final answer is:
\[ \frac{y + 1}{x} = -\frac{1}{2} \]