Answer :
To determine whether a linear equation has infinitely many solutions, we look at equations of the form
[tex]$$
ax+b=cx+d.
$$[/tex]
Such an equation has infinitely many solutions if and only if the coefficients of [tex]\( x \)[/tex] and the constant terms on both sides are the same, that is, if
[tex]$$
a = c \quad \text{and} \quad b = d.
$$[/tex]
Let's analyze each option:
1. Option A:
The equation is
[tex]$$
-76x + 76 = 76x + 76.
$$[/tex]
Here, the coefficient on the left is [tex]\(-76\)[/tex] and on the right is [tex]\(76\)[/tex]. Since [tex]\(-76 \neq 76\)[/tex], the condition is not met. Thus, the equation does not have infinitely many solutions.
2. Option B:
The equation is
[tex]$$
-76x + 76 = -76x + 76.
$$[/tex]
In this case, the coefficient on both sides is [tex]\(-76\)[/tex] and the constant on both sides is [tex]\(76\)[/tex]. Since both conditions [tex]\( -76 = -76 \)[/tex] and [tex]\( 76 = 76 \)[/tex] hold, the equation has infinitely many solutions.
3. Option C:
The equation is
[tex]$$
76x+76 = -76x+76.
$$[/tex]
The coefficients are [tex]\(76\)[/tex] (left) and [tex]\(-76\)[/tex] (right). Since these coefficients are not equal, the equation does not have infinitely many solutions.
4. Option D:
The equation is
[tex]$$
76x+76 = 76x+76.
$$[/tex]
Here, both sides have the same coefficient [tex]\(76\)[/tex] and the same constant [tex]\(76\)[/tex]. Thus, the equation has infinitely many solutions.
Based on our analysis, the equations in Option B and Option D have infinitely many solutions.
[tex]$$
ax+b=cx+d.
$$[/tex]
Such an equation has infinitely many solutions if and only if the coefficients of [tex]\( x \)[/tex] and the constant terms on both sides are the same, that is, if
[tex]$$
a = c \quad \text{and} \quad b = d.
$$[/tex]
Let's analyze each option:
1. Option A:
The equation is
[tex]$$
-76x + 76 = 76x + 76.
$$[/tex]
Here, the coefficient on the left is [tex]\(-76\)[/tex] and on the right is [tex]\(76\)[/tex]. Since [tex]\(-76 \neq 76\)[/tex], the condition is not met. Thus, the equation does not have infinitely many solutions.
2. Option B:
The equation is
[tex]$$
-76x + 76 = -76x + 76.
$$[/tex]
In this case, the coefficient on both sides is [tex]\(-76\)[/tex] and the constant on both sides is [tex]\(76\)[/tex]. Since both conditions [tex]\( -76 = -76 \)[/tex] and [tex]\( 76 = 76 \)[/tex] hold, the equation has infinitely many solutions.
3. Option C:
The equation is
[tex]$$
76x+76 = -76x+76.
$$[/tex]
The coefficients are [tex]\(76\)[/tex] (left) and [tex]\(-76\)[/tex] (right). Since these coefficients are not equal, the equation does not have infinitely many solutions.
4. Option D:
The equation is
[tex]$$
76x+76 = 76x+76.
$$[/tex]
Here, both sides have the same coefficient [tex]\(76\)[/tex] and the same constant [tex]\(76\)[/tex]. Thus, the equation has infinitely many solutions.
Based on our analysis, the equations in Option B and Option D have infinitely many solutions.
