Question

In the equation 4x+1=−ax–4, where a is a constant, which of the following values of a results in an equation with exactly one solution?

a) 4
b) -4
c) Neither value
d) Both values

Answer 1

For the equation 4x + 1 = -ax - 4 to have exactly one solution, 'a' cannot be -4 or 4, as these would lead to either no solutions or infinitely many solutions, respectively. Thus, the correct answer is (c) Neither value.

Step-by-step explanation:

The equation 4x + 1 = -ax - 4 presents a situation where we are trying to determine a value for 'a' that will result in only one solution to the equation. The equation is currently a linear equation, not a quadratic equation. To have exactly one solution, the coefficients of 'x' on both sides of the equation should not be equal; otherwise, the equation would either have no solutions or infinitely many solutions (in the case where the two expressions are identical).

Combining like terms, we get (4 + a)x = -5. For there to be exactly one solution, 'a' cannot be -4 as this would negate the x term entirely (making the equation 0x = -5, which has no solution) and cannot be 4 as this would make both sides equal after distributing (resulting in an identity 0 = 0, with infinitely many solutions). Therefore, neither a = 4 nor a = -4 will result in an equation with exactly one solution, which makes our answer (c) Neither value.