What two equations have infinite solutions?
2x+3=2x+7
7x+3=7x+3
4x+22=8x+10
5x+5=5x+4
3x+18=5x+8
3(4x+8)=4(3x+6)
check your work
The first equation, 2x+3=2x+7, has no solutions since the variable x is eliminated when subtracting 2x from both sides. Therefore, this equation does not have infinite solutions.
The second equation, 7x+3=7x+3, is an identity. It means that both sides of the equation are equal for every value of x. Therefore, this equation has infinite solutions.
The third equation, 4x+22=8x+10, has one solution. By simplifying and solving for x, we find x = 3.
The fourth equation, 5x+5=5x+4, has no solutions since the variable x is eliminated when subtracting 5x from both sides. Therefore, this equation does not have infinite solutions.
The fifth equation, 3x+18=5x+8, has one solution. By simplifying and solving for x, we find x = 5.
The sixth equation, 3(4x+8)=4(3x+6), can be simplified to 12x+24=12x+24. This is an identity, similar to the second equation. Therefore, this equation has infinite solutions.
Checking the work, we find that the second and sixth equations have infinite solutions.
What two have no solutions?
2x+3=2x+7
7x+3=7x+3
4x+22=8x+10
5x+5=5x+4
3x+18=5x+8
3(4x+8)=4(3x+6)
Check your work.
Apologies for the previous mistakes. Let's re-evaluate the equations:
The first equation, 2x+3=2x+7, simplifies to 0=4, which is not true. Therefore, this equation has no solutions.
The second equation, 7x+3=7x+3, is an identity. It means that both sides of the equation are equal for every value of x. Therefore, this equation has infinite solutions.
The third equation, 4x+22=8x+10, simplifies to 0=12, which is not true. Therefore, this equation has no solutions.
The fourth equation, 5x+5=5x+4, simplifies to 1=4, which is not true. Therefore, this equation has no solutions.
The fifth equation, 3x+18=5x+8, can be simplified to -2 = 2x. Solving for x, we get x = -1. Therefore, this equation has a single solution, not no solutions.
The sixth equation, 3(4x+8)=4(3x+6), can be further simplified to 12x + 24 = 12x + 24. This is another identity, similar to the second equation. Therefore, this equation has infinite solutions.
To summarize, the first, third, and fourth equations have no solutions, while the second and sixth equations have infinite solutions. The fifth equation has a single solution.
To check if the given equations have infinite solutions, we need to determine if the equations are equivalent or not.
1. For the equation 2x+3=2x+7:
Subtracting 2x from both sides, we get 3=7.
This is not possible since 3 does not equal 7.
Therefore, this equation does not have infinite solutions.
2. For the equation 7x+3=7x+3:
Subtracting 7x from both sides, we get 3=3.
This equation is true, and the variables cancel out.
Therefore, this equation has infinite solutions.
3. For the equation 4x+22=8x+10:
Subtracting 4x from both sides, we get 22=4x+10.
Subtracting 10 from both sides, we get 12=4x.
Dividing both sides by 4, we get 3=x.
This equation has a specific solution for x.
Therefore, this equation does not have infinite solutions.
4. For the equation 5x+5=5x+4:
Subtracting 5x from both sides, we get 5=4.
This is not possible since 5 does not equal 4.
Therefore, this equation does not have infinite solutions.
5. For the equation 3x+18=5x+8:
Subtracting 3x from both sides, we get 18=2x+8.
Subtracting 8 from both sides, we get 10=2x.
Dividing both sides by 2, we get 5=x.
This equation has a specific solution for x.
Therefore, this equation does not have infinite solutions.
6. For the equation 3(4x+8)=4(3x+6):
Distributing on both sides, we get 12x+24=12x+24.
Subtracting 12x from both sides and subtracting 24 from both sides, we get 0=0.
This equation is true, and the variables cancel out.
Therefore, this equation has infinite solutions.
Based on the above steps, the equations that have infinite solutions are:
- 7x+3=7x+3
- 3(4x+8)=4(3x+6)
To determine which equations have infinite solutions, we need to look for equations where both sides of the equation are identical or equivalent. Let's analyze each equation:
1. 2x+3=2x+7:
To check if this equation has infinite solutions, we can subtract 2x from both sides to isolate the constant term:
2x - 2x + 3 = 2x - 2x + 7
0x + 3 = 0x + 7
We have 3 = 7, which is not true. Therefore, this equation does not have infinite solutions.
2. 7x+3=7x+3:
Here, both sides of the equation are identical. It means that no matter what value x takes, the equation will always hold true. Thus, this equation has infinite solutions.
3. 4x+22=8x+10:
To check this equation, we need to simplify both sides. First, we can start by subtracting 4x from both sides:
4x - 4x + 22 = 8x - 4x + 10
0x + 22 = 4x + 10
22 = 4x + 10
Next, we subtract 10 from both sides:
22 - 10 = 4x + 10 - 10
12 = 4x
Dividing both sides by 4:
12/4 = 4x/4
3 = x
Thus, this equation has a single solution, x = 3, rather than infinite solutions.
4. 5x+5=5x+4:
Similar to the previous example, let's subtract 5x from both sides:
5x - 5x + 5 = 5x - 5x + 4
0x + 5 = 0x + 4
We have 5 = 4, which is not true. Therefore, this equation does not have infinite solutions.
5. 3x+18=5x+8:
Starting by subtracting 3x from both sides:
3x - 3x + 18 = 5x - 3x + 8
0x + 18 = 2x + 8
Next, subtracting 18 from both sides:
18 - 18 = 2x + 8 - 18
0 = 2x - 10
Rearranging the equation:
0 = 2(x - 5)
This equation implies that for any value of x, the left side will always be zero, while the right side will always be zero when x = 5. Therefore, this equation does not have infinite solutions.
6. 3(4x+8)=4(3x+6):
We can start by distributing the 3 on the left side and the 4 on the right side:
12x + 24 = 12x + 24
Here, both sides of the equation are identical. Thus, this equation has infinite solutions.
Checking the work:
To confirm the results, we can solve each equation to verify if the obtained values satisfy the original equation. For example, in equation 2, 7x+3=7x+3, we could choose any value for x and substitute it back into the equation to confirm that it holds true.
If you provide the specific equation you want us to check, I can go through the verification process and confirm the results for you.