Which equations have infinitely many solutions?
6(x-9)=6x-54+x
5(x-8)+10=5(x+2)
20-8x-6=2(7-4x)
4(x-+4)=2(2x+5)+6
The equation "4(x-6)=2(2x+5)+6" has infinitely many solutions.
is there another one.
Apologies for the confusion. Let me correct myself.
The equation "4(x-6)=2(2x+5)+6" does not have infinitely many solutions.
The equation "5(x-8)+10=5(x+2)" has infinitely many solutions.
To determine which equations have infinitely many solutions, we need to simplify and compare the expressions.
1. 6(x-9)=6x-54+x
Let's distribute 6 to (x-9):
6x - 54 = 6x - 54 + x
The x terms cancel out when we combine like terms:
-54 = -54
The equation simplifies to -54 = -54. This is true for any value of x, so the equation has infinitely many solutions.
2. 5(x-8)+10=5(x+2)
Let's distribute 5 to (x-8):
5x - 40 + 10 = 5x + 10
Combine like terms:
5x - 30 = 5x + 10
The x terms cancel out when we subtract 5x from both sides:
-30 = 10
The equation simplifies to -30 = 10 which is not true. Therefore, this equation has no solution.
3. 20-8x-6=2(7-4x)
Simplify the equation:
14 - 8x = 14 - 8x
The equation simplifies to 14 - 8x = 14 - 8x. This is true for any value of x, so the equation has infinitely many solutions.
4. 4(x-4)=2(2x+5)+6
Let's distribute 4 to (x-4) and 2 to (2x+5):
4x - 16 = 4x + 10 + 6
Combine like terms:
4x - 16 = 4x + 16
The x terms cancel out when we subtract 4x from both sides:
-16 = 16
The equation simplifies to -16 = 16 which is not true. Therefore, this equation has no solution.
To summarize,
The equations that have infinitely many solutions are:
1. 6(x-9) = 6x - 54 + x
3. 20 - 8x - 6 = 2(7 - 4x)
To determine which equations have infinitely many solutions, we need to solve the equations and see if they simplify to an identity (true for all values) or a contradiction (false for all values).
1. 6(x-9) = 6x-54+x
To solve this equation, we will distribute the 6 on the left side:
6x - 54 = 6x - 54 + x
Combining like terms, we have:
6x - 54 = 7x - 54
We can see that the variables cancel out, and we are left with:
-54 = -54
This is an identity, meaning that it holds true for all values of x. Therefore, the equation has infinitely many solutions.
2. 5(x-8) + 10 = 5(x+2)
Again, we will distribute the 5 on the left side:
5x - 40 + 10 = 5x + 10
Combining like terms:
5x - 30 = 5x + 10
Here, we can see that the variable terms cancel out, leaving us with:
-30 = 10
This is a contradiction, which means the equation has no solution. There are no values of x that satisfy the equation.
3. 20 - 8x - 6 = 2(7 - 4x)
Start by simplifying the right side:
20 - 8x - 6 = 14 - 8x
Combine like terms on the left side:
14 - 8x = 14 - 8x
Once again, the variable terms cancel out leaving us with:
14 = 14
This is an identity, so the equation has infinitely many solutions.
4. 4(x - 4) = 2(2x + 5) + 6
First, simplify the right side:
4x - 16 = 4x + 10 + 6
Combine like terms:
4x - 16 = 4x + 16
In this case, the equation has no variable terms remaining after combining like terms:
-16 = 16
This is a contradiction, meaning the equation has no solution.
To summarize:
- The first equation, 6(x-9) = 6x-54+x, and the third equation, 20 - 8x - 6 = 2(7 - 4x), both have infinitely many solutions.
- The second equation, 5(x-8) + 10 = 5(x+2), and the fourth equation, 4(x - 4) = 2(2x + 5) + 6, have no solutions.