Which equations have infinitely many solutions? Select TWO that apply.
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
2x+9=18+2x
The equations that have infinitely many solutions are:
5(x-8)+10=5(x+2)
2x+9=18+2x
To determine which equations have infinitely many solutions, we can simplify the equations and observe if any variables cancel out. Let's go through each equation:
1. 6(x-9) = 6x-54+x
Expanding the left side gives us:
6x - 54 = 6x - 54 + x
Simplifying both sides yields:
6x - 54 = 7x - 54
By subtracting 6x from both sides, we have:
-54 = x - 54
If we subtract 54 from both sides, we get:
-54 - (-54) = x - 54 - (-54)
Simplifying:
0 = x - 0
This equation simplifies to:
0 = x
From this, we see that x can take any value, since any number subtracted by zero is itself. Therefore, the equation has infinitely many solutions.
2. 5(x-8) + 10 = 5(x+2)
Expanding and simplifying both sides gives us:
5x - 40 + 10 = 5x + 10
Combining like terms:
5x - 30 = 5x + 10
Subtracting 5x from both sides:
-30 = 10
This equation is not true since -30 does not equal 10. Therefore, this equation has no solution.
3. 20 - 8x - 6 = 2(7 - 4x)
Simplifying both sides gives us:
14 - 8x = 14 - 8x
If we subtract 14 from both sides, we get:
-8x = -8x
Here, every term cancels out, resulting in an identity. Therefore, this equation also has infinitely many solutions.
4. 4(x+4) = 2(2x+5) + 6
Expanding and simplifying both sides gives us:
4x + 16 = 4x + 10 + 6
Combining like terms:
4x + 16 = 4x + 16
Subtracting 4x from both sides:
16 = 16
This equation is a true statement, meaning that both sides are equal. Therefore, this equation has infinitely many solutions.
5. 2x + 9 = 18 + 2x
If we subtract 2x from both sides, we get:
9 = 18
This equation is not true since 9 does not equal 18. Therefore, this equation has no solution.
In summary, 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 check if any variables cancel out when we simplify the equations.
Let's analyze each equation:
1) 6(x-9) = 6x-54+x
To simplify, distribute the 6:
6x - 54 = 6x - 54 + x
If we combine like terms, we end up with:
6x - 54 = 7x - 54
Here, we observe that the variable "x" cancels out on both sides of the equation. This means that regardless of the value of "x," the equation will always be true. Hence, this equation has infinitely many solutions.
2) 5(x-8) + 10 = 5(x+2)
Again, let's distribute the 5:
5x - 40 + 10 = 5x + 10
If we simplify, we get:
5x - 30 = 5x + 10
Notice that the variable "x" appears on both sides of the equation. However, after simplifying, we see that the equation becomes:
-30 = 10
This is an inconsistent equation since -30 is not equal to 10. Therefore, this equation has no solution.
3) 20 - 8x - 6 = 2(7 - 4x)
Simplifying by distributing the 2 on the right:
20 - 8x - 6 = 14 - 8x
Combine like terms:
14 - 8x = 14 - 8x
Here, we see that the variable "x" cancels out on both sides of the equation. As a result, this equation has infinitely many solutions.
4) 4(x+4) = 2(2x+5) + 6
Simplifying:
4x + 16 = 4x + 10 + 6
Combining like terms:
4x + 16 = 4x + 16
Again, we observe that the variable "x" cancels out on both sides of the equation. This implies that this equation has infinitely many solutions.
5) 2x + 9 = 18 + 2x
By simplifying, we find:
2x + 9 = 2x + 18
Here, notice that we have the same variable on both sides, but after simplifying:
9 = 18
Since 9 is not equal to 18, this equation has no solution.
In summary, the equations that have infinitely many solutions are:
- 6(x-9)=6x-54+x
- 20-8x-6=2(7-4x)