Which equation has no solution? Select all that apply.
(1 point)
Responses
8k+38=−3(−6−4k)
8 k plus 38 is equal to negative 3 times open paren negative 6 minus 4 k close paren
−2(w−2)=−3−2w
negative 2 times open paren w minus 2 close paren is equal to negative 3 minus 2 w
−4(x+3)=−12−4x
negative 4 times open paren x plus 3 close paren is equal to negative 12 minus 4 x
36−7y=−7(y−5)
The equation that has no solution is:
−2(w−2)=−3−2w
This equation leads to a contradiction and cannot be solved.
To determine which equation has no solution, we need to solve each equation and see if we find a valid solution.
1. 8k + 38 = -3 (-6 - 4k)
Simplifying both sides, we get 8k + 38 = 18 + 12k
Rearranging the equation, we get 8k - 12k = 18 - 38
Simplifying further, -4k = -20
Dividing by -4, we get k = 5
This equation has a valid solution, so it is not the one with no solution.
2. -2(w - 2) = -3 - 2w
Expanding the equation, we get -2w + 4 = -3 - 2w
Simplifying both sides, we get -2w + 2w = -3 - 4
0 = -7
This equation leads to an inconsistency, where 0 is not equal to -7. Therefore, this equation has no solution.
3. -4(x + 3) = -12 - 4x
Expanding the equation, we get -4x - 12 = -12 - 4x
Simplifying both sides, we get -4x + 4x = -12 + 12
0 = 0
This equation leads to an identity, where 0 is equal to 0. Therefore, this equation has infinitely many solutions, not no solution.
4. 36 - 7y = -7(y - 5)
Expanding the equation, we get 36 - 7y = -7y + 35
Simplifying both sides, we get 36 - 35 = -7y + 7y
1 = 0
This equation leads to an inconsistency, where 1 is not equal to 0. Therefore, this equation has no solution.
The equation that has no solution is: -2(w-2) = -3 - 2w
To determine which equation(s) have no solution, we need to solve each equation and see if we end up with a logical contradiction, such as a statement like 2 = 3 or 0 = 1.
Let's solve each equation and check if it leads to a contradiction:
1) 8k + 38 = -3(-6 - 4k):
First, distribute -3 to (-6 - 4k): -3 * -6 = 18 and -3 * -4k = 12k. The equation becomes: 8k + 38 = 18 + 12k.
Combine like terms: 38 = 18 + 4k. Subtract 18 from both sides: 20 = 4k.
Divide both sides by 4: 5 = k.
This equation has a unique solution, so it is not the correct answer.
2) -2(w - 2) = -3 - 2w:
First, distribute -2 to (w - 2): -2w + 4 = -3 - 2w.
Notice that the variable terms (-2w and -2w) cancel out on both sides, resulting in a constant equation: 4 = -3.
This is a contradiction; 4 cannot be equal to -3. Therefore, this equation has no solution.
3) -4(x + 3) = -12 - 4x:
First, distribute -4 to (x + 3): -4x - 12 = -12 - 4x.
The variable terms (-4x and -4x) cancel out on both sides, resulting in a true statement: -12 = -12.
This is a consistent equation; -12 is indeed equal to -12. Therefore, this equation has infinite solutions, not no solution.
4) 36 - 7y = -7(y - 5):
First, distribute -7 to (y - 5): 36 - 7y = -7y + 35.
Move all terms involving y to one side: 36 = 35.
This is a contradiction; 36 cannot be equal to 35. Therefore, this equation has no solution.
Therefore, the equations that have no solution are:
-2(w - 2) = -3 - 2w
36 - 7y = -7(y - 5)
So, the correct answer is:
-2(w - 2) = -3 - 2w
36 - 7y = -7(y - 5)