Let's solve the problem step-by-step by comparing each given expression with the expression formed by subtracting the sum of two numbers [tex]\( r \)[/tex] and [tex]\( s \)[/tex] from twice the difference of [tex]\( r \)[/tex] minus [tex]\( s \)[/tex].
First, let's write down the expression given in the problem:
[tex]\[ 2(r - s) - (r + s) \][/tex]
Now, we need to compare this expression with each of the four given options to find the equivalent one.
### Option 1: [tex]\( 2(r - s) - r + s \)[/tex]
Let's simplify [tex]\( 2(r - s) - r + s \)[/tex]:
[tex]\[ 2(r - s) - r + s = 2r - 2s - r + s = r - s \][/tex]
### Option 2: [tex]\( 2r - s - r + s \)[/tex]
Let's simplify [tex]\( 2r - s - r + s \)[/tex]:
[tex]\[ 2r - s - r + s = 2r - r = r \][/tex]
### Option 3: [tex]\( 2r - s - (r + s) \)[/tex]
Let's simplify [tex]\( 2r - s - (r + s) \)[/tex]:
[tex]\[ 2r - s - (r + s) = 2r - s - r - s = r - 2s \][/tex]
### Option 4: [tex]\( 2(r - s) - (r + 2) \)[/tex]
Let's simplify [tex]\( 2(r - s) - (r + 2) \)[/tex]:
[tex]\[ 2(r - s) - (r + 2) = 2r - 2s - r - 2 = r - 2s - 2\][/tex]
### Original Expression:
[tex]\[ 2(r - s) - (r + s) \][/tex]
Let's simplify this:
[tex]\[ 2(r - s) - (r + s) = 2r - 2s - r - s = r - 3s \][/tex]
Comparing all four options with the simplified form [tex]\( r - 3s \)[/tex], we find that:
- Option 1 simplifies to [tex]\( r - s \)[/tex]
- Option 2 simplifies to [tex]\( r \)[/tex]
- Option 3 simplifies to [tex]\( 2r - r - s - s = r - 3s \)[/tex]
- Option 4 simplifies to [tex]\( r - 2s - 2 \)[/tex]
Therefore, the expression equivalent to [tex]\( 2(r - s) - (r + s) \)[/tex] is [tex]\( 2r - s - (r + s) \)[/tex], which matches option 3.
Thus, the correct answer is: [tex]\( 2r - s - (r + s) \)[/tex]
[tex]\(\boxed{3}\)[/tex]
