To determine which equation is equivalent to the given equation [tex]\( 7m + 11 = -4(2m + 3) \)[/tex], we should solve for [tex]\( m \)[/tex]. Let's go through the steps of solving this equation:
1. Expand and Simplify:
Begin by expanding the right-hand side of the equation:
[tex]\[
7m + 11 = -4(2m + 3)
\][/tex]
[tex]\[
7m + 11 = -4 \cdot 2m - 4 \cdot 3
\][/tex]
[tex]\[
7m + 11 = -8m - 12
\][/tex]
2. Isolate the variable [tex]\( m \)[/tex] on one side:
Add [tex]\( 8m \)[/tex] to both sides to combine the terms involving [tex]\( m \)[/tex]:
[tex]\[
7m + 8m + 11 = -12
\][/tex]
[tex]\[
15m + 11 = -12
\][/tex]
3. Move the constant term to the other side:
Subtract 11 from both sides to isolate the term with [tex]\( m \)[/tex]:
[tex]\[
15m = -12 - 11
\][/tex]
[tex]\[
15m = -23
\][/tex]
4. Solve for [tex]\( m \)[/tex]:
Divide both sides by 15:
[tex]\[
m = \frac{-23}{15}
\][/tex]
Now, we have the value of [tex]\( m \)[/tex]. To find the equivalent equation, we can compare the provided options with our steps.
1. Option A: [tex]\( -m = 1 \)[/tex]
[tex]\[
m = -1
\][/tex]
This is not equivalent since [tex]\( m \neq -1 \)[/tex].
2. Option B: [tex]\( -15m = -23 \)[/tex]
[tex]\[
-15 \cdot \frac{-23}{15} = -23
\][/tex]
Simplifies to [tex]\( 15m = -23 \)[/tex], which contains the same coefficient structure as our derived equation.
3. Option C: [tex]\( -m = -1 \)[/tex]
[tex]\[
m = 1
\][/tex]
This is not equivalent since [tex]\( m \neq 1 \)[/tex].
4. Option D: [tex]\( 15m = -23 \)[/tex]
[tex]\[
This is exactly the equation we derived.
\][/tex]
Based on our solution, the correct answer is:
[tex]\[
\boxed{15m = -23}
\][/tex]
Therefore, the equation:
[tex]\[
15m = -23
\][/tex]
is equivalent to the given equation. Hence, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
