Complete the equation so that it has infinite solutions.(1 point)

Question

Complete the equation so that it has infinite solutions.(1 point)
4x+10= (response)x + (response)

GPT-4o mini · Accepted Answer

To make the equation $4x + 10 = (response)x + (response)$ have infinite solutions, the two sides of the equation must be equivalent for all values of $x$. This means that the coefficients of $x$ and the constant terms must be the same on both sides. 1. The coefficient of $x$ on the left side is 4. Thus, the coefficient on the right side (response) should also be 4. 2. The constant on the left side is 10. Therefore, the constant on the right side (response) must also be 10. So, we can complete the equation as follows: $$ 4x + 10 = 4x + 10 $$ Therefore, the responses are both 4 for the $x$ coefficient and 10 for the constant term: - Response for $x$: 4 - Response for constant: 10 Thus, the completed equation is: $$ 4x + 10 = 4x + 10 $$

BTS ARMY · Answer

Complete the equation so that it has no solution.(1 point) 5x+12= (response)x+9

GPT-4o mini · Answer

To make the equation $5x + 12 = (response)x + 9$ have no solutions, we need to ensure that both sides of the equation are parallel, which means the coefficients of $x$ must be the same, but the constant terms must be different. 1. The coefficient of $x$ on the left side is 5, so we set the $x$ coefficient on the right side (response) to 5. 2. The constant on the left side is 12, and we need the constant on the right side to be a different value than 9 to create a contradiction. We can choose any value other than 9 for the right side's constant term. For example, 10 would work. So, we can complete the equation as follows: $$ 5x + 12 = 5x + 10 $$ Here, the response for $x$ is 5, and the constant on the right side is 10. Thus, the completed equation is: $$ 5x + 12 = 5x + 10 $$ This equation has no solutions since it simplifies to $12 = 10$, which is a false statement.

BTS ARMY · Answer

Complete the equation so that it has no solution.(1 point) −7x−12= (response)x+12

GPT-4o mini · Answer

To make the equation $-7x - 12 = (response)x + 12$ have no solutions, we need to ensure that the coefficients of $x$ on both sides are the same, but the constant terms are different. This creates a situation where the lines are parallel. 1. The coefficient of $x$ on the left side is $-7$. Therefore, we set the coefficient on the right side (response) to $-7$. 2. The constant term on the left side is $-12$, and we need the constant on the right side (response) to be different from $12$. A good choice here would be $13$ or any value other than $12$. So, we can complete the equation as follows: $$ -7x - 12 = -7x + 13 $$ Thus, the final equation is: $$ -7x - 12 = -7x + 13 $$ This equation has no solutions because it simplifies to $-12 = 13$, which is a contradiction.

BTS ARMY · Answer

Solve for x
to determine if the equation has one, none, or infinite solutions.

6(2x+4)−24=12x

(1 point)
Responses

The equation has one solution: x=−20
.
The equation has one solution: x equals negative 20 .

The equation has one solution: x=0
.
The equation has one solution: x equals 0 .

The equation has no solution.
The equation has no solution.

The equation has infinite solutions.

GPT-4o mini · Answer

To solve the equation $6(2x + 4) - 24 = 12x$, let's first simplify the left side: 1. Distribute the 6: $$ 6(2x) + 6(4) - 24 = 12x $$ which simplifies to: $$ 12x + 24 - 24 = 12x $$ thus resulting in: $$ 12x = 12x $$ 2. Now, if we subtract $12x$ from both sides: $$ 12x - 12x = 12x - 12x $$ which simplifies to: $$ 0 = 0 $$ Since we end up with the statement $0 = 0$, this indicates that the equation is true for all values of $x$. Therefore, the equation has **infinite solutions**. The correct response is: - The equation has infinite solutions.

BTS ARMY · Answer

Solve for x
to determine if the equation has one, none, or infinite solutions.

9(x+1)=3(5x+1)−12

(1 point)
Responses

The equation has no solution.
The equation has no solution.

The equation has one solution: x=0
.
The equation has one solution: x equals 20 .

The equation has infinite solutions.
The equation has infinite solutions.

The equation has one solution: x=3
.

GPT-4o mini · Answer

To solve the equation $9(x + 1) = 3(5x + 1) - 12$, we will first simplify both sides. 1. **Distribute on the left side:** $$ 9(x + 1) = 9x + 9 $$ 2. **Distribute on the right side:** $$ 3(5x + 1) - 12 = 15x + 3 - 12 = 15x - 9 $$ Now, the equation looks like this: $$ 9x + 9 = 15x - 9 $$ 3. **Rearranging the equation:** To move all terms involving $x$ to one side and the constants to the other side, we can subtract $9x$ from both sides: $$ 9 = 15x - 9x - 9 $$ This simplifies to: $$ 9 = 6x - 9 $$ 4. **Add 9 to both sides:** $$ 9 + 9 = 6x $$ which simplifies to: $$ 18 = 6x $$ 5. **Dividing by 6:** $$ x = \frac{18}{6} = 3 $$ Now that we have solved for $x$, we find that: - The equation has **one solution: $x = 3$**. Thus, the correct response is: - The equation has one solution: $x = 3$.

BTS ARMY · Answer

Complete the equation so that it has no solution. −7x−12= (response)x+12

GPT-4o mini · Answer

To make the equation $-7x - 12 = (response)x + 12$ have no solutions, we need the coefficients of $x$ to be the same on both sides but the constant terms to be different. 1. The coefficient of $x$ on the left side is $-7$. Therefore, we set the coefficient on the right side (response) to $-7$. 2. The constant term on the left side is $-12$, and the constant term on the right side is $12$. To create a situation with no solutions, we need to ensure the constants are different. Since the right side already has $12$, we will keep it as is to meet this condition. Putting this together, we get: $$ -7x - 12 = -7x + 12 $$ So, the completed equation that has no solution is: $$ -7x - 12 = -7x + 12 $$ This equation has no solutions because it simplifies to $-12 = 12$, which is a contradiction.