To determine which equation is equivalent to [tex]\(3(a+4) - 5(a-2) = 22\)[/tex], we need to first simplify the given equation step-by-step. Let's start by expanding and simplifying the terms.
Given the equation:
[tex]\[ 3(a + 4) - 5(a - 2) = 22 \][/tex]
Step 1: Expand the terms inside the parentheses.
[tex]\[ 3(a + 4) - 5(a - 2) \][/tex]
[tex]\[ = 3a + 12 - 5a + 10 \][/tex]
Step 2: Combine like terms.
[tex]\[ = (3a - 5a) + (12 + 10) \][/tex]
[tex]\[ = -2a + 22 \][/tex]
So, we have:
[tex]\[ -2a + 22 = 22 \][/tex]
Step 3: Isolate [tex]\(a\)[/tex] on one side of the equation.
Subtract 22 from both sides:
[tex]\[ -2a + 22 - 22 = 22 - 22 \][/tex]
[tex]\[ -2a = 0 \][/tex]
Thus, all terms involving [tex]\(a\)[/tex] cancel out in this equation, leading to zeroes on both sides.
Given the options:
A. [tex]\(8a + 2 = 22\)[/tex]
B. [tex]\(8a + 22 = 22\)[/tex]
C. [tex]\(-2a + 2 = 22\)[/tex]
D. [tex]\(-2a + 22 = 22\)[/tex]
Step 4: Check if any of the given equations are equivalent to the simplified form [tex]\(-2a + 22 = 22\)[/tex].
1. Option A: [tex]\(8a + 2 = 22\)[/tex]
Subtract 2 from both sides:
[tex]\[ 8a + 2 - 2 = 22 - 2 \][/tex]
[tex]\[ 8a = 20 \][/tex]
The resulting simplified equation is:
[tex]\[ 8a = 20 \][/tex]
This equation does not simplify to [tex]\(-2a + 22 = 22\)[/tex].
2. Option B: [tex]\(8a + 22 = 22\)[/tex]
Subtract 22 from both sides:
[tex]\[ 8a + 22 - 22 = 22 - 22 \][/tex]
[tex]\[ 8a = 0 \][/tex]
The resulting simplified equation is:
[tex]\[ 8a = 0 \][/tex]
This equation does not simplify to [tex]\(-2a + 22 = 22\)[/tex].
3. Option C: [tex]\(-2a + 2 = 22\)[/tex]
Subtract 2 from both sides:
[tex]\[ -2a + 2 - 2 = 22 - 2 \][/tex]
[tex]\[ -2a = 20 \][/tex]
The resulting simplified equation is:
[tex]\[ -2a = 20 \][/tex]
This equation does not simplify to [tex]\(-2a + 22 = 22\)[/tex].
4. Option D: [tex]\(-2a + 22 = 22\)[/tex]
This is already in the form [tex]\(-2a + 22 = 22\)[/tex].
Upon comparing each step, we conclude that the only equation equivalent to the original given equation is:
[tex]\[ \boxed{-2a + 22 = 22} \][/tex]
