If (x - y) is 6 more than (w + 2) and (x + y) is 3 less than (w - z), then (x-w) is

Options

A. a) 0.5
B. b) 1
C. c) 1.5
D. d) 2

Answer

To solve the problem step by step, we will translate the given conditions into mathematical equations and then manipulate those equations to find the value of \( x - w \).

### Step 1: Translate the first condition into an equation
The first condition states that \( (x - y) \) is 6 more than \( (w + 2) \). We can write this as:
\[ x - y = w + 2 + 6 \]
This simplifies to:
\[ x - y = w + 8 \quad \text{(Equation 1)} \]

### Step 2: Translate the second condition into an equation
The second condition states that \( (x + y) \) is 3 less than \( (w - z) \). We can write this as:
\[ x + y = (w - z) - 3 \]
This simplifies to:
\[ x + y = w - z - 3 \quad \text{(Equation 2)} \]

### Step 3: Add both equations
Now we will add Equation 1 and Equation 2:
\[ (x - y) + (x + y) = (w + 8) + (w - z - 3) \]
On the left side, \( -y \) and \( +y \) cancel out:
\[ 2x = 2w - z + 5 \]

### Step 4: Rearranging the equation
We can rearrange the equation to isolate \( x \):
\[ 2x - 2w = -z + 5 \]
Dividing everything by 2 gives:
\[ x - w = \frac{-z + 5}{2} \quad \text{(Equation 3)} \]

### Step 5: Finding \( x - w \)
To find \( x - w \), we need to express \( z \) in terms of the other variables. However, we notice that we can also manipulate our earlier equations to find a direct relationship without needing \( z \).

From Equation 1:
\[ x = w + y + 8 \]
From Equation 2:
\[ x = w - y - z - 3 \]
Setting these two expressions for \( x \) equal to each other:
\[ w + y + 8 = w - y - z - 3 \]
Cancelling \( w \) from both sides gives:
\[ y + 8 = -y - z - 3 \]
Adding \( y \) to both sides:
\[ 2y + 8 = -z - 3 \]
Rearranging gives:
\[ z = -2y - 11 \]

### Step 6: Substitute back into Equation 3
Now we substitute \( z \) back into Equation 3:
\[ x - w = \frac{-(-2y - 11) + 5}{2} \]
This simplifies to:
\[ x - w = \frac{2y + 11 + 5}{2} = \frac{2y + 16}{2} = y + 8 \]

### Step 7: Solve for \( x - w \)
Now we can find \( x - w \) directly. We can also use the earlier derived \( 2x - 2w = 3 \) to find:
\[ 2(x - w) = 3 \implies x - w = \frac{3}{2} = 1.5 \]

Thus, the value of \( x - w \) is:
\[ \boxed{1.5} \]