Which of the following is an example of an equation with no solution?
A. 4x+3=3x+3
B. 3x+4=3x+4
C. 3x+4=3x+3
D. 3x+4=4x+3
The correct answer is C. 3x+4=3x+3. This equation can be rewritten as 4=3, which is not true. Thus, there is no value of x that satisfies this equation, making it a contradiction and having no solution.
Which of the following is an example of an equation with exactly one solution?
A. 3x+4=3x+3
B. 3x+4=4x+3
C. 3x+4=3x+4
D. 4x+3=4x+3
The correct answer is D. 4x+3=4x+3. This equation can be simplified to 0=0, which is always true regardless of the value of x. Since the equation is always true, it has an infinite number of solutions. Thus, there is exactly one solution for this equation.
Solve for x to determine if the equation has one, none, or infinite solutions.
3(x+2)−6=3x.
A. The equation has one solution: x=3.
B. The equation has no solution.
C. The equation has infinite solutions.
D. The equation has one solution: x=−3.
Let's solve for x and determine if the equation has one, none, or infinite solutions:
3(x+2) - 6 = 3x
Distributing the 3 on the left side of the equation:
3x + 6 - 6 = 3x
Combining like terms:
3x = 3x
This equation states that the left side of the equation is equal to the right side for any value of x. Therefore, it is an identity and has infinite solutions.
The correct answer is C. The equation has infinite solutions.
Solve for x to determine if the equation has one, none, or infinite solutions.
4x=3(x+5)−15.
A. The equation has one solution: x=−10.
B. The equation has no solution.
C. The equation has infinite solutions.
D. The equation has one solution: x=0.
Let's solve for x to determine if the equation has one, none, or infinite solutions:
4x = 3(x+5) - 15
First, distribute on the right side of the equation:
4x = 3x + 15 - 15
Combine like terms on the right side:
4x = 3x
The x term is the same on both sides of the equation. We can subtract 3x from both sides to get:
4x - 3x = 3x - 3x
Simplifying,
x = 0
From this solution, we can see that there is one unique solution.
The correct answer is D. The equation has one solution: x = 0.
Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)−7x=3x+6.
A. The equation has one solution: x=6.
B. The equation has one solution: x=10.
C. The equation has infinite solutions.
D. The equation has no solution.
Let's solve for x to determine if the equation has one, none, or infinite solutions:
5(2x+2) - 7x = 3x + 6
First, distribute on the left side of the equation:
10x + 10 - 7x = 3x + 6
Combine like terms on the left side:
3x + 10 = 3x + 6
Now, subtract 3x from both sides:
10 = 6
This equation states that 10 is equal to 6, which is not true. Therefore, there is no value of x that satisfies this equation.
The correct answer is D. The equation has no solution.