To determine which equations have the same value of [tex]\( x \)[/tex] as the given equation [tex]\(\frac{2}{3}(6x + 12) = -24 \)[/tex], we need to follow these steps:
1. Solve the given equation [tex]\(\frac{2}{3}(6x + 12) = -24\)[/tex] to find the value of [tex]\( x \)[/tex]:
[tex]\[
\frac{2}{3}(6x + 12) = -24
\][/tex]
First, eliminate the fraction by multiplying both sides by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
6x + 12 = -24 \times \frac{3}{2}
\][/tex]
Simplify the right side:
[tex]\[
6x + 12 = -36
\][/tex]
Next, solve for [tex]\( x \)[/tex] by isolating the variable:
[tex]\[
6x = -36 - 12
\][/tex]
[tex]\[
6x = -48
\][/tex]
[tex]\[
x = \frac{-48}{6}
\][/tex]
[tex]\[
x = -8
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( -8 \)[/tex].
2. Check each proposed equation to see if they yield the same value of [tex]\( x \)[/tex]:
Let's analyze each equation one by one and see if they provide [tex]\( x = -8 \)[/tex].
#### Option 1: [tex]\( 4x + 8 = -24 \)[/tex]
Substitute [tex]\( x = -8 \)[/tex]:
[tex]\[
4(-8) + 8 = -32 + 8 = -24
\][/tex]
This is a correct equation, as it holds true for [tex]\( x = -8 \)[/tex].
#### Option 2: [tex]\( 9x + 18 = -24 \)[/tex]
Substitute [tex]\( x = -8 \)[/tex]:
[tex]\[
9(-8) + 18 = -72 + 18 = -54
\][/tex]
This equation does NOT hold true for [tex]\( x = -8 \)[/tex].
#### Option 3: [tex]\( 4x = -16 \)[/tex]
Substitute [tex]\( x = -8 \)[/tex]:
[tex]\[
4(-8) = -32
\][/tex]
This equation does NOT hold true for [tex]\( x = -8 \)[/tex].
#### Option 4: [tex]\( \frac{18x + 36}{2} = -24 \)[/tex]
Substitute [tex]\( x = -8 \)[/tex]:
[tex]\[
\frac{18(-8) + 36}{2} = \frac{-144 + 36}{2} = \frac{-108}{2} = -54
\][/tex]
This equation does NOT hold true for [tex]\( x = -8 \)[/tex].
#### Option 5: [tex]\( 4x = -32 \)[/tex]
Substitute [tex]\( x = -8 \)[/tex]:
[tex]\[
4(-8) = -32
\][/tex]
This is a correct equation, as it holds true for [tex]\( x = -8 \)[/tex].
Therefore, the two equations that have the same value of [tex]\( x \)[/tex] as the given equation [tex]\(\frac{2}{3}(6x + 12) = -24\)[/tex] are:
- [tex]\( 4x + 8 = -24 \)[/tex]
- [tex]\( 4x = -32 \)[/tex]
These correspond to options 1 and 5.
