Solve the equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. (4x)/(x-3)=(12)/(x-3)+7
The Correct Answer and Explanation is:
Let’s solve the equation step by step:
Given: 4xx−3=12x−3+7\frac{4x}{x-3} = \frac{12}{x-3} + 7x−34x=x−312+7
Step 1: Eliminate the denominator
Since both sides have a denominator of x−3x – 3x−3, we can multiply both sides of the equation by x−3x – 3x−3, assuming x≠3x \neq 3x=3 to avoid division by zero. 4x=12+7(x−3)4x = 12 + 7(x – 3)4x=12+7(x−3)
Step 2: Simplify the equation
Expand the right-hand side: 4x=12+7(x−3)4x = 12 + 7(x – 3)4x=12+7(x−3)
Distribute the 7: 4x=12+7x−214x = 12 + 7x – 214x=12+7x−21
Now, simplify the constants: 4x=7x−94x = 7x – 94x=7x−9
Step 3: Move terms involving xxx to one side
Subtract 7x7x7x from both sides: 4x−7x=−94x – 7x = -94x−7x=−9
Simplify: −3x=−9-3x = -9−3x=−9
Step 4: Solve for xxx
Divide both sides by -3: x=3x = 3x=3
Step 5: Check for extraneous solutions
We need to check if x=3x = 3x=3 is valid in the original equation. If x=3x = 3x=3, the denominators in the original equation would become zero, which would make the equation undefined.
Therefore, x=3x = 3x=3 is not a valid solution.
Conclusion
Since the solution leads to an invalid value (because it makes the denominator zero), the equation has no solution. This type of equation is known as an inconsistent equation because there is no value for xxx that satisfies it.
In summary:
- The equation has no solution.
- It is an inconsistent equation.
