Determine whether the equation below is an identity or whether it has no solution.
-6(3x+7)=-18x+42
-6(3x+7)=-18x+42 can be simplified as follows:
-18x - 42 = -18x + 42
When we subtract -18x from both sides, we get:
-42 = 42
Since -42 is not equal to 42, the equation has no solution.
To determine whether the equation -6(3x+7)=-18x+42 is an identity or has no solution, we need to simplify the equation and compare the coefficients of x on both sides.
First, distribute -6 to the terms inside the parentheses:
-6 * 3x + -6 * 7 = -18x + 42
This simplifies to:
-18x - 42 = -18x + 42
Next, we will rearrange the equation by subtracting -18x from both sides:
-18x - 18x - 42 = -18x - 18x + 42
This simplifies to:
-36x - 42 = 42 - 18x
Now, combine like terms:
-36x - 42 = 42 - 18x
Rearrange the equation by adding 18x to both sides:
-36x + 18x - 42 = 42 - 18x + 18x
This simplifies to:
-18x - 42 = 42
Next, add 42 to both sides:
-18x - 42 + 42 = 42 + 42
This simplifies to:
-18x = 84
Finally, to find the value of x, divide both sides of the equation by -18:
-18x / -18 = 84 / -18
This simplifies to:
x = -84/18
Further simplifying gives:
x = -14/3
Since we have found a numerical solution for x, the equation does have a solution. Therefore, the equation -6(3x+7)=-18x+42 is not an identity, but rather has a solution of x = -14/3.