Which of the equations has one solution? Select all that apply.
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-2(w +2) = -3 - 2w
8k + 38 = -3(-6 - 4k)
-4(x + 3) = -12 - 4x
36 - 7y = -7(y - 5)
-2(w +2) = -3 - 2w
-4(x + 3) = -12 - 4x
To determine which equations have one solution, we can simplify each equation and see if there are any variables that cancel out. Let's go through each equation:
1. -2(w + 2) = -3 - 2w:
Starting with the left side:
-2(w + 2) = -2w - 4
Now we can compare the left side (-2w - 4) with the right side (-3 - 2w). Since they both have the same terms (-2w) and the same constant (-4 and -3 are both negative 3 in value), this equation has one solution.
2. 8k + 38 = -3(-6 - 4k):
Simplifying the right side:
-3(-6 - 4k) = 18 + 12k
Now we can compare the left side (8k + 38) with the right side (18 + 12k). Since they have different terms, this equation does not have one solution.
3. -4(x + 3) = -12 - 4x:
Expanding the left side:
-4(x + 3) = -4x - 12
Now we can compare the left side (-4x - 12) with the right side (-12 - 4x). Since they have the same terms (-4x) and the same constant (-12), this equation has one solution.
4. 36 - 7y = -7(y - 5):
Applying the distributive property on the right side:
-7(y - 5) = -7y + 35
Now we can compare the left side (36 - 7y) with the right side (-7y + 35). Since they have different terms, this equation does not have one solution.
So the equations that have one solution are:
-2(w + 2) = -3 - 2w
-4(x + 3) = -12 - 4x
To determine which equations have one solution, we need to simplify each equation and see if there is one unique value that satisfies the equation. Let's analyze each equation separately:
1. -2(w + 2) = -3 - 2w
Distribute the -2 on the left side:
-2w - 4 = -3 - 2w
Combine like terms:
-4 = -3
This equation is inconsistent because it does not have a valid solution. It does not have one solution.
2. 8k + 38 = -3(-6 - 4k)
Distribute the -3 on the right side:
8k + 38 = 18 + 12k
Combine like terms:
8k + 38 = 18 + 12k
Subtracting 8k from both sides:
38 = 18 + 4k
Subtracting 18 from both sides:
20 = 4k
Dividing both sides by 4:
5 = k
This equation has one solution, k = 5.
3. -4(x + 3) = -12 - 4x
Distribute the -4 on the left side:
-4x - 12 = -12 - 4x
Combine like terms:
-12 = -12
This equation is always true. It means that any value of x will satisfy this equation. Therefore, it has infinite solutions, not one solution.
4. 36 - 7y = -7(y - 5)
Distribute the -7 on the right side:
36 - 7y = -7y + 35
Add 7y to both sides:
36 = 35
This equation is inconsistent because it does not have a valid solution. It does not have one solution.
In conclusion, the equations that have one solution are:
-2(w + 2) = -3 - 2w
8k + 38 = -3(-6 - 4k)
Please note that the third equation and the fourth equation have infinite solutions or are inconsistent, respectively.