Which equation has exactly ONE solution?
A. 8+3x/2 = 13 + x
B. 3+12x/2 = 6x +15
C. 10+15x/5 = 3x + 10
D. 4x-20/4 = x - 5
To determine which equation has exactly one solution, we can calculate the number of solutions by simplifying each equation and observing the relationship between the variables.
A. 8+3x/2 = 13 + x
To remove the fractions, we can multiply both sides of the equation by 2:
2(8 + 3x/2) = 2(13 + x)
16 + 3x = 26 + 2x
Simplifying the equation gives:
3x - 2x = 26 - 16
x = 10
B. 3+12x/2 = 6x +15
Simplifying the equation gives:
3 + 6x = 6x +15
The variable terms cancel out, leaving:
3 = 15
This equation has no solution.
C. 10+15x/5 = 3x + 10
To remove the fraction, we can multiply both sides of the equation by 5:
5(10 + 15x/5) = 5(3x + 10)
50 + 15x = 15x + 50
The variable terms cancel out, leaving:
50 = 50
This equation has an infinite number of solutions.
D. 4x-20/4 = x - 5
Simplifying the equation gives:
x - 5 = x - 5
The variable terms cancel out, leaving:
-5 = -5
This equation also has an infinite number of solutions.
Therefore, the equation that has exactly one solution is A. 8+3x/2 = 13 + x.
To determine which equation has exactly one solution, we need to simplify each equation and analyze the variables.
Let's go through each option:
A. 8 + 3x/2 = 13 + x
To simplify this, let's get rid of the fractions by multiplying each term by 2:
16 + 3x = 26 + 2x
Moving all the x terms to one side:
3x - 2x = 26 - 16
x = 10
So, option A has a unique solution.
B. 3 + 12x/2 = 6x + 15
Again, let's eliminate the fraction by multiplying each term by 2:
6 + 12x = 12x + 30
Notice that both terms contain the variable x and they are equal, which means the equation simplifies to:
6 = 30
This is a contradiction, so option B does not have a solution.
C. 10 + 15x/5 = 3x + 10
Let's simplify the fraction:
10 + 3x = 3x + 10
Here, all the terms are equal, which means there are infinitely many solutions. This type of equation is called an identity.
So, option C does not have only one solution.
D. 4x - 20/4 = x - 5
Let's simplify the fraction first:
4x - 5 = x - 5
Here, the variables on both sides cancel out, resulting in:
0 = 0
This equation is also an identity, meaning it has infinitely many solutions.
So, option D does not have only one solution.
In conclusion, the equation from option A, 8 + 3x/2 = 13 + x, has exactly one solution.
To determine which equation has exactly one solution, we need to simplify each equation and see if we can isolate the variable on one side of the equation.
Let's go through each equation:
A. 8+3x/2 = 13 + x
To simplify this equation, we can start by multiplying both sides of the equation by 2 to eliminate the fraction:
2 * (8 + 3x/2) = 2 * (13 + x)
16 + 3x = 26 + 2x
Next, let's combine like terms by subtracting 2x from both sides:
16 + 3x - 2x = 26 + 2x - 2x
16 + x = 26
Finally, let's isolate the variable by subtracting 16 from both sides:
16 + x - 16 = 26 - 16
x = 10
So, the solution to equation A is x = 10.
B. 3+12x/2 = 6x + 15
To simplify this equation, we can start by multiplying both sides of the equation by 2 to eliminate the fraction:
2 * (3 + 12x/2) = 2 * (6x + 15)
6 + 12x = 12x + 30
Notice that the variable "x" cancels out on both sides of the equation. This means that the equation is an identity, and it is true for any value of x. Therefore, equation B does not have exactly one solution.
C. 10+15x/5 = 3x + 10
To simplify this equation, we can start by multiplying both sides of the equation by 5 to eliminate the fraction:
5 * (10 + 15x/5) = 5 * (3x + 10)
50 + 15x = 15x + 50
Notice that after simplifying, both sides of the equation are equal. This means that the equation is an identity, and it is true for any value of x. Therefore, equation C does not have exactly one solution.
D. 4x-20/4 = x - 5
To simplify this equation, we can start by multiplying both sides of the equation by 4 to eliminate the fraction:
4 * (4x - 20/4) = 4 * (x - 5)
16x - 20 = 4x - 20
Notice that after simplifying, both sides of the equation are equal. This means that the equation is an identity, and it is true for any value of x. Therefore, equation D does not have exactly one solution.
Out of the given equations, only equation A has exactly one solution, which is x = 10.