Let's solve the given equation step-by-step to determine which expression is equivalent to [tex]\( 8(2x - 5) - 4(3x - 7) = 7 \)[/tex].
First, we need to distribute the numbers outside the parentheses through each term inside the parentheses.
### Step 1: Distribute the 8 in [tex]\( 8(2x - 5) \)[/tex]
[tex]\[
8(2x - 5) = 8 \cdot 2x - 8 \cdot 5 = 16x - 40
\][/tex]
### Step 2: Distribute the -4 in [tex]\( -4(3x - 7) \)[/tex]
[tex]\[
-4(3x - 7) = -4 \cdot 3x + 4 \cdot 7 = -12x + 28
\][/tex]
### Step 3: Combine the results from both distributions
[tex]\[
8(2x - 5) - 4(3x - 7) = 16x - 40 - 12x + 28
\][/tex]
### Step 4: Simplify the combined expression
Combine like terms:
[tex]\[
(16x - 12x) + (-40 + 28) = 4x - 12
\][/tex]
So, the simplified form of the left-hand side of the equation is [tex]\( 4x - 12 \)[/tex]. Setting this equal to the right-hand side, we get:
[tex]\[
4x - 12 = 7
\][/tex]
### Conclusion
Given these steps, let's see which of the provided options matches the equivalent expression:
1. [tex]\( 16x - 40 - 12x + 28 = 7 \)[/tex]
2. [tex]\( 16x - 5 - 12x - 7 = 7 \)[/tex]
3. [tex]\( 16x - 40 - 12x - 28 = 7 \)[/tex]
4. [tex]\( 16x - 5 + 12 - 7 = 7 \)[/tex]
From our steps, the correct equivalent expression is:
[tex]\[
16x - 40 - 12x + 28 = 7
\][/tex]
Hence, the equivalent expression to [tex]\( 8(2x - 5) - 4(3x - 7) = 7 \)[/tex] is:
[tex]\[
16x - 40 - 12x + 28 = 7
\][/tex]
