Question

Which equation is an identity?


9 – (2v + 3) = –2v – 6


7m – 5 = 8m + 7 – m


–3y + 3 = –3y – 6


3w + 8 – w = 4w – 2(w – 4)

Answer 1

To determine which equation is an identity (an equation that is true for all values of the variable), we can simplify each equation.

1. **Equation 1:**
\[
9 - (2v + 3) = -2v - 6
\]
Simplifying the left side:
\[
9 - 2v - 3 = -2v - 6 \implies 6 - 2v = -2v - 6
\]
Now, if we add \(2v\) to both sides:
\[
6 = -6
\]
This is not true, so this equation is not an identity.

2. **Equation 2:**
\[
7m - 5 = 8m + 7 - m
\]
Simplifying the right side:
\[
8m - m + 7 = 7m + 7
\]
So, the equation becomes:
\[
7m - 5 = 7m + 7
\]
Now, if we subtract \(7m\) from both sides:
\[
-5 = 7
\]
This is not true, so this equation is not an identity.

3. **Equation 3:**
\[
-3y + 3 = -3y - 6
\]
We can add \(3y\) to both sides:
\[
3 = -6
\]
This is not true, so this equation is not an identity.

4. **Equation 4:**
\[
3w + 8 - w = 4w - 2(w - 4)
\]
Simplifying the left side:
\[
(3w - w) + 8 = 2w + 8
\]
For the right side:
\[
4w - 2w + 8 = 2w + 8
\]
So, it simplifies to:
\[
2w + 8 = 2w + 8
\]
This is true for all values of \(w\) and represents an identity.

Thus, **the fourth equation \(3w + 8 - w = 4w - 2(w - 4)\) is an identity.**