Which equation has no solution?
A. 5v + 3 = 5v - 5
B. 2x - 4 = 2x + 7 - x
C.2y + 5 - y = 2y - 3(y - 6)
D.7z + 9 = -7z - 6
The equation that has no solution is D. 7z + 9 = -7z - 6.
To determine which equation has no solution, we need to check if the variables cancel out on both sides of the equation, leading to a contradiction.
Let's go through each option one by one:
A. 5v + 3 = 5v - 5
In this equation, we can simplify by subtracting both sides by 5v:
3 = -5
However, we see that this leads to a contradiction since 3 is not equal to -5. Therefore, option A has no solution.
B. 2x - 4 = 2x + 7 - x
Here, we can simplify the equation by combining like terms:
2x - 4 = x + 7
Next, we can rearrange the equation to isolate the variables on one side:
2x - x = 7 + 4
x = 11
By solving the equation, we find that x has a specific value. Therefore, option B does have a solution.
C. 2y + 5 - y = 2y - 3(y - 6)
To solve this equation, we will apply the distributive property:
2y + 5 - y = 2y - 3y + 18
Next, we simplify by combining like terms:
y + 5 = -y + 18
To isolate the variable y, we can add y to both sides:
y + y + 5 = -y + y + 18
2y + 5 = 18
Then, we subtract 5 from both sides to isolate 2y:
2y + 5 - 5 = 18 - 5
2y = 13
Finally, dividing both sides by 2 gives us:
y = 6.5
Since y has a specific value, option C does have a solution.
D. 7z + 9 = -7z - 6
Let's simplify the equation by adding 7z to both sides:
7z + 7z + 9 = -7z + 7z - 6
14z + 9 = -6
Now, subtracting 9 from both sides, we have:
14z + 9 - 9 = -6 - 9
14z = -15
Finally, dividing both sides by 14 gives us:
z = -15/14
Since z has a specific value, option D does have a solution.
In summary, the equation that has no solution is option A: 5v + 3 = 5v - 5.
To find which equation has no solution, we need to determine if any equation leads to a contradiction or an inconsistency. Let's analyze each equation step by step:
A. 5v + 3 = 5v - 5
To isolate the variable v, we subtract 5v from both sides: 3 = -5. This is a contradiction since 3 does not equal -5. Thus, this equation has no solution.
B. 2x - 4 = 2x + 7 - x
Simplifying both sides of the equation, we have 2x - 4 = 2x + 7 - x.
Combining like terms, we get 2x - 4 = x + 7.
Next, we subtract x from both sides: x - 4 = 7.
Now, we add 4 to both sides: x = 11.
Therefore, x = 11 is the solution to this equation. Thus, this equation has a solution.
C. 2y + 5 - y = 2y - 3(y - 6)
Let's simplify both sides of the equation step by step.
Starting with the left side of the equation:
2y + 5 - y simplifies to y + 5.
Now, let's simplify the right side of the equation:
2y - 3(y - 6) Using the distributive property, we get 2y - 3y + 18.
This simplifies to -y + 18.
Now, our equation becomes y + 5 = -y + 18.
Let's isolate the variable y. Subtracting y from both sides of the equation, we get 2y = 13.
Finally, dividing both sides by 2, we find y = 6.5. Thus, this equation has a solution.
D. 7z + 9 = -7z - 6
Let's simplify both sides of the equation step by step.
Starting with the left side of the equation:
7z + 9
Now, let's simplify the right side of the equation:
-7z - 6
Combining like terms, we have 7z + 9 = -7z - 6.
To isolate the variable z, we can add 7z to both sides of the equation: 14z + 9 = -6.
Subtracting 9 from both sides, we get 14z = -15.
Finally, dividing both sides by 14, we find z = -15/14. Thus, this equation has a solution.
In summary, the equation that has no solution is A. 5v + 3 = 5v - 5.