Which of the following is an example of an equation with no solution?(1 point)
Responses
3x+4=4x+3
3 x plus 4 equals 4 x plus 3
3x+4=3x+4
3 x plus 4 equals 3 x plus 4
4x+3=3x+3
4 x plus 3 equals 3 x plus 3
3x+4=3x+3
3x+4=3x+3 is an example of an equation with no solution.
Which of the following is an example of an equation with exactly one solution?(1 point)
Responses
3x+4=3x+4
3 x plus 4 equals 3 x plus 4
3x+4=4x+3
3 x plus 4 equals 4 x plus 3
3x+4=3x+3
3 x plus 4 equals 3 x plus 3
4x+3=4x+3
4x+3=4x+3 is an example of an equation with exactly one solution.
Justify why −12(x−3)−52=−16 has one solution, infinitely many solutions, or no solution.(1 point)
Responses
It has one solution, because the solution is in the form x=a. The solution is x=−12.
It has one solution, because the solution is in the form x equals a . The solution is x equals negative 12 .
It has no solution because the solution is in the form a=b.
It has no solution because the solution is in the form a equals b .
It has infinitely many solutions because the solution is in the form a=a.
It has infinitely many solutions because the solution is in the form a equals a .
It has one solution because the solution is in the form x=a. The solution is x=0.
In order to justify whether the equation −12(x−3)−52=−16 has one solution, infinitely many solutions, or no solution, we need to consider the operations involved and simplify the equation.
Starting with the given equation:
−12(x−3)−52=−16
First, distribute the -12 to the terms inside the parentheses:
-12x + 36 - 52 = -16
Combine like terms:
-12x - 16 = -16
Next, isolate the variable on one side of the equation by adding 16 to both sides:
-12x = 0
Finally, divide both sides by -12 to solve for x:
x = 0
Therefore, the equation has one solution, x = 0. Hence, the correct response is:
It has one solution because the solution is in the form x=a. The solution is x=0.
Solve for x to determine if the equation has one, none, or infinite solutions.
3(x+2)−6=3x(1 point)
Responses
The equation has no solution.
The equation has no solution.
The equation has one solution: x=3.
The equation has one solution: x equals 3 .
The equation has infinite solutions.
The equation has infinite solutions.
The equation has one solution: x=−3.
To determine if the equation 3(x+2)−6=3x has one, none, or infinite solutions, we can simplify and solve for x.
Starting with the given equation:
3(x+2)−6=3x
First, distribute the 3 to the terms inside the parentheses:
3x + 6 - 6 = 3x
Combine like terms:
3x = 3x
The equation simplifies to 3x = 3x. In this case, both sides of the equation are equal, indicating that the equation is an identity and not a conditional statement. Therefore, it has infinite solutions.
Thus, the correct response is: The equation has infinite solutions.
Solve for x to determine if the equation has one, none, or infinite solutions.
4x=3(x+5)−15(1 point)
Responses
The equation has one solution: x=0.
The equation has one solution: x equals 0 .
The equation has infinite solutions.
The equation has infinite solutions.
The equation has no solution.
The equation has no solution.
The equation has one solution: x=−10.
To determine if the equation 4x=3(x+5)−15 has one, none, or infinite solutions, we can simplify and solve for x.
Starting with the given equation:
4x = 3(x+5)−15
First, distribute the 3 to the terms inside the parentheses:
4x = 3x + 15 - 15
Combine like terms:
4x = 3x
Next, subtract 3x from both sides to isolate the variable:
4x - 3x = 0
Simplifying the left side gives:
x = 0
Therefore, the equation has one solution: x = 0.
Thus, the correct response is: The equation has one solution: x=0.