Answer :
Sure! Let's examine each of the equations to determine which have infinitely many solutions.
Equation A: [tex]\( 76x + 76 = -76x + 76 \)[/tex]
1. Move all [tex]\( x \)[/tex]-terms to one side:
[tex]\[
76x + 76x + 76 = 76
\][/tex]
2. Combine like terms:
[tex]\[
152x + 76 = 76
\][/tex]
3. Subtract 76 from both sides:
[tex]\[
152x = 0
\][/tex]
4. Divide by 152:
[tex]\[
x = 0
\][/tex]
This equation has exactly one solution, [tex]\( x = 0 \)[/tex], not infinitely many.
Equation B: [tex]\( 76x + 76 = 76x + 76 \)[/tex]
1. Subtract [tex]\( 76x \)[/tex] from both sides:
[tex]\[
76 = 76
\][/tex]
This is a true statement, which is an identity, and holds for all [tex]\( x \)[/tex]. Therefore, this equation has infinitely many solutions.
Equation C: [tex]\( -76x + 76 = -76x + 76 \)[/tex]
1. Subtract [tex]\(-76x\)[/tex] from both sides:
[tex]\[
76 = 76
\][/tex]
This is also a true statement, an identity, holding for all [tex]\( x \)[/tex]. Hence, this equation has infinitely many solutions.
Equation D: [tex]\( -76x + 76 = 76x + 76 \)[/tex]
1. Move all [tex]\( x \)[/tex]-terms to one side:
[tex]\[
-76x - 76x + 76 = 76
\][/tex]
2. Combine like terms:
[tex]\[
-152x + 76 = 76
\][/tex]
3. Subtract 76 from both sides:
[tex]\[
-152x = 0
\][/tex]
4. Divide by [tex]\(-152\)[/tex]:
[tex]\[
x = 0
\][/tex]
This equation also has exactly one solution, [tex]\( x = 0 \)[/tex], not infinitely many.
Conclusion:
- Equation B and Equation C have infinitely many solutions.
- Equations A and D do not have infinitely many solutions.
Equation A: [tex]\( 76x + 76 = -76x + 76 \)[/tex]
1. Move all [tex]\( x \)[/tex]-terms to one side:
[tex]\[
76x + 76x + 76 = 76
\][/tex]
2. Combine like terms:
[tex]\[
152x + 76 = 76
\][/tex]
3. Subtract 76 from both sides:
[tex]\[
152x = 0
\][/tex]
4. Divide by 152:
[tex]\[
x = 0
\][/tex]
This equation has exactly one solution, [tex]\( x = 0 \)[/tex], not infinitely many.
Equation B: [tex]\( 76x + 76 = 76x + 76 \)[/tex]
1. Subtract [tex]\( 76x \)[/tex] from both sides:
[tex]\[
76 = 76
\][/tex]
This is a true statement, which is an identity, and holds for all [tex]\( x \)[/tex]. Therefore, this equation has infinitely many solutions.
Equation C: [tex]\( -76x + 76 = -76x + 76 \)[/tex]
1. Subtract [tex]\(-76x\)[/tex] from both sides:
[tex]\[
76 = 76
\][/tex]
This is also a true statement, an identity, holding for all [tex]\( x \)[/tex]. Hence, this equation has infinitely many solutions.
Equation D: [tex]\( -76x + 76 = 76x + 76 \)[/tex]
1. Move all [tex]\( x \)[/tex]-terms to one side:
[tex]\[
-76x - 76x + 76 = 76
\][/tex]
2. Combine like terms:
[tex]\[
-152x + 76 = 76
\][/tex]
3. Subtract 76 from both sides:
[tex]\[
-152x = 0
\][/tex]
4. Divide by [tex]\(-152\)[/tex]:
[tex]\[
x = 0
\][/tex]
This equation also has exactly one solution, [tex]\( x = 0 \)[/tex], not infinitely many.
Conclusion:
- Equation B and Equation C have infinitely many solutions.
- Equations A and D do not have infinitely many solutions.
