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Answer :
To determine which of the given equations have infinitely many solutions, let's analyze each one step-by-step.
Equation A: [tex]\(-76x + 76 = 76x + 76\)[/tex]
1. Start by trying to simplify both sides of the equation.
2. Subtract [tex]\(76x\)[/tex] from both sides:
[tex]\[
-76x + 76 - 76x = 76x + 76 - 76x
\][/tex]
Simplifying, we get:
[tex]\[
-152x + 76 = 76
\][/tex]
3. Subtract 76 from both sides:
[tex]\[
-152x + 76 - 76 = 76 - 76
\][/tex]
Which simplifies to:
[tex]\[
-152x = 0
\][/tex]
4. Divide both sides by [tex]\(-152\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 0
\][/tex]
This equation has one solution: [tex]\(x = 0\)[/tex].
Equation B: [tex]\(76x + 76 = 76x + 76\)[/tex]
1. Notice that both sides of the equation are identical, meaning:
[tex]\[
76x + 76 = 76x + 76
\][/tex]
2. This means every [tex]\(x\)[/tex] value you choose will satisfy the equation because both sides are equal.
This equation has infinitely many solutions.
Equation C: [tex]\(-76x + 76 = -76x + 76\)[/tex]
1. Notice that both sides are identical, meaning:
[tex]\[
-76x + 76 = -76x + 76
\][/tex]
2. This means every [tex]\(x\)[/tex] value you choose will satisfy the equation because both sides are equal.
This equation also has infinitely many solutions.
Equation D: [tex]\(76x + 76 = -76x + 76\)[/tex]
1. Start by trying to simplify both sides of the equation.
2. Subtract [tex]\(76\)[/tex] from both sides:
[tex]\[
76x + 76 - 76 = -76x + 76 - 76
\][/tex]
Simplifying, we get:
[tex]\[
76x = -76x
\][/tex]
3. Add [tex]\(76x\)[/tex] to both sides to get:
[tex]\[
76x + 76x = 0
\][/tex]
[tex]\[
152x = 0
\][/tex]
4. Divide both sides by 152 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 0
\][/tex]
This equation has one solution: [tex]\(x = 0\)[/tex].
Conclusion
Equations B and C have infinitely many solutions.
Equation A: [tex]\(-76x + 76 = 76x + 76\)[/tex]
1. Start by trying to simplify both sides of the equation.
2. Subtract [tex]\(76x\)[/tex] from both sides:
[tex]\[
-76x + 76 - 76x = 76x + 76 - 76x
\][/tex]
Simplifying, we get:
[tex]\[
-152x + 76 = 76
\][/tex]
3. Subtract 76 from both sides:
[tex]\[
-152x + 76 - 76 = 76 - 76
\][/tex]
Which simplifies to:
[tex]\[
-152x = 0
\][/tex]
4. Divide both sides by [tex]\(-152\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 0
\][/tex]
This equation has one solution: [tex]\(x = 0\)[/tex].
Equation B: [tex]\(76x + 76 = 76x + 76\)[/tex]
1. Notice that both sides of the equation are identical, meaning:
[tex]\[
76x + 76 = 76x + 76
\][/tex]
2. This means every [tex]\(x\)[/tex] value you choose will satisfy the equation because both sides are equal.
This equation has infinitely many solutions.
Equation C: [tex]\(-76x + 76 = -76x + 76\)[/tex]
1. Notice that both sides are identical, meaning:
[tex]\[
-76x + 76 = -76x + 76
\][/tex]
2. This means every [tex]\(x\)[/tex] value you choose will satisfy the equation because both sides are equal.
This equation also has infinitely many solutions.
Equation D: [tex]\(76x + 76 = -76x + 76\)[/tex]
1. Start by trying to simplify both sides of the equation.
2. Subtract [tex]\(76\)[/tex] from both sides:
[tex]\[
76x + 76 - 76 = -76x + 76 - 76
\][/tex]
Simplifying, we get:
[tex]\[
76x = -76x
\][/tex]
3. Add [tex]\(76x\)[/tex] to both sides to get:
[tex]\[
76x + 76x = 0
\][/tex]
[tex]\[
152x = 0
\][/tex]
4. Divide both sides by 152 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 0
\][/tex]
This equation has one solution: [tex]\(x = 0\)[/tex].
Conclusion
Equations B and C have infinitely many solutions.
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