Let's analyze whether any of the equations are equivalent by solving them step-by-step to check for their solutions.
1. Equation 1: [tex]\( 2x + 7 = 1 \)[/tex]
Simplifying this equation, we get:
[tex]\[
2x + 7 = 1
\][/tex]
Subtracting 7 from both sides:
[tex]\[
2x = 1 - 7
\][/tex]
[tex]\[
2x = -6
\][/tex]
Dividing both sides by 2:
[tex]\[
x = -3
\][/tex]
2. Equation 2: [tex]\( -3x + 4 = 12 \)[/tex]
Simplifying this equation, we get:
[tex]\[
-3x + 4 = 12
\][/tex]
Subtracting 4 from both sides:
[tex]\[
-3x = 12 - 4
\][/tex]
[tex]\[
-3x = 8
\][/tex]
Dividing both sides by -3:
[tex]\[
x = \frac{8}{-3}
\][/tex]
[tex]\[
x = -\frac{8}{3}
\][/tex]
3. Equation 3: [tex]\( x = -3 \)[/tex]
This equation is already simplified:
[tex]\[
x = -3
\][/tex]
Now we compare the solutions:
- Equation [tex]\( 2x + 7 = 1 \)[/tex] has the solution [tex]\( x = -3 \)[/tex].
- Equation [tex]\( -3x + 4 = 12 \)[/tex] has the solution [tex]\( x = -\frac{8}{3} \)[/tex].
- Equation [tex]\( x = -3 \)[/tex] is already in the form [tex]\( x = -3 \)[/tex].
From the above analysis, we see that the solutions for the equations [tex]\( 2x + 7 = 1 \)[/tex] and [tex]\( x = -3 \)[/tex] are both [tex]\( x = -3 \)[/tex]. Therefore, these two equations are equivalent.
The correct statement should be:
"The equations [tex]\( 2x + 7 = 1 \)[/tex] and [tex]\( x = -3 \)[/tex] are equivalent because both have the same solution, -3."
