Which of the following equations has exactly one solution?(1 point)
Responses
−3x+8=−3x−8
negative 3 x plus 8 equals negative 3 x minus 8
−3x−8=3x−8
negative 3 x minus 8 equals 3 x minus 8
−8x+3=−8x+3
negative 8 x plus 3 equals negative 8 x plus 3
3x−8=3x+8
The equation −3x+8=−3x−8 has exactly one solution.
To find the equation that has exactly one solution, we need to determine which equation does not result in any cancelation of variables or coefficients.
Let's examine each equation:
1) −3x+8=−3x−8
If we simplify this equation, we will cancel out the variable x:
-3x + 8 = -3x - 8
Subtracting -3x from both sides:
8 = -8
This equation has no solution.
2) −3x−8=3x−8
If we simplify this equation, we will cancel out the variable x:
-3x - 8 = 3x - 8
Adding 3x and 8 to both sides:
0 = 0
This equation has infinite solutions, not exactly one.
3) −8x+3=−8x+3
If we simplify this equation, we will not cancel out any variables or coefficients:
-8x + 3 = -8x + 3
This equation has infinitely many solutions, not exactly one.
4) 3x−8=3x+8
If we simplify this equation, we will have:
3x - 8 = 3x + 8
Subtracting 3x from both sides:
-8 = 8
This equation also has no solution.
Therefore, none of the given equations has exactly one solution.
To determine which equation has exactly one solution, we need to compare the coefficients and constants in each equation.
Let's analyze each equation:
1) −3x+8=−3x−8
This equation has the same coefficients (-3) for the x term on both sides, but different constants (8 and -8) on each side. When we simplify the equation, we get 8 = -8, which is not true. Therefore, this equation has no solution.
2) −3x−8=3x−8
This equation has the same coefficient (-3) for the x term on both sides, and the same constant (-8) on each side. When we simplify the equation, we get -3x - 8 = 3x - 8, which simplifies to -3x = 3x. This means that the x term cancels out, and we are left with 0 = 0. Since this is a true statement, it means that this equation has infinitely many solutions.
3) −8x+3=−8x+3
This equation has the same coefficients (-8) for the x term on both sides, and the same constant (3) on each side. When we simplify the equation, we get -8x + 3 = -8x + 3, which simplifies to -8x = -8x. Again, the x term cancels out, and we are left with 0 = 0. This means that this equation also has infinitely many solutions.
4) 3x−8=3x+8
This equation has the same coefficient (3) for the x term on both sides, but different constants (-8 and 8) on each side. When we simplify the equation, we get 3x - 8 = 3x + 8, which simplifies to -8 = 8. Since this is not a true statement, it means that this equation has no solution.
Out of the given equations, only equation 2) −3x−8=3x−8 has infinitely many solutions. The other equations either have no solution or contradict themselves.