(a - 12)x = a + 8. In the equation above, a is a constant. If the equation has no solutions, what is the value of a?
Answer (2)
The value of a for which the equation ( a − 12 ) x = a + 8 has no solutions is 12 . This is because if a = 12 , the equation results in an undefined expression. Thus, no values of x can satisfy the equation.
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To solve for the value of the constant a in the equation ( a − 12 ) x = a + 8 such that the equation has no solutions, we need to understand when a linear equation has no solutions.
A linear equation of the form a x + b = 0 typically has no solutions if it simplifies to a contradiction, such as 0 = 1 . This happens when the coefficients of x on both sides of the equation are equal and the constant terms are different.
Let's break down the steps:
Identify the Expression : We have ( a − 12 ) x = a + 8 .
Condition for No Solution : For the equation to have no solutions, the coefficient of x on the left side must equal the coefficient of x on the right side, and the constant terms must differ.
Equalize Coefficients : In this equation, the right side has no x term, so its coefficient is 0. For the left side, the coefficient of x is a − 12 .
For the coefficients to be equal, we must have: a − 12 = 0 This results in: a = 12
Check Constant Condition : Substitute a = 12 back into the equation, ( 12 − 12 ) x = 12 + 8 Simplifying gives us: 0 ⋅ x = 20
This equation simplifies to 0 = 20 , which is a contradiction.
Therefore, the value of a that makes the equation have no solutions is a = 12 . This results in the equation becoming a contradiction, confirming that a = 12 is indeed the correct answer for no solutions.
