Question

What values of the variable make a true equality?

A) (x^2 + x) / x = x + 1

B) (3y^2 + 4y + 1) / (y+1) = 3y + 1

C) (x^2 - 4x - 5) / (x - 5) = x + 1

D) (3a^2 - ac + 2c - 6a) / (3a - c) = a - 2

Answer 1

The values that make a true equality for each equation are: A) all values of x, B) all values of y, C) all values of x, D) all values of a.

Step-by-step explanation:

The values of the variable that make a true equality can be found by solving each equation.

A) For the equation (x^2 + x) / x = x + 1, we simplify the left side since x in the denominator cancels out, resulting in (x+1) = x+1, which is true for all values of x.

B) For the equation (3y^2 + 4y + 1) / (y+1) = 3y + 1, we simplify the left side by dividing each term by (y+1), resulting in 3y+1 = 3y+1, which is again true for all values of y.

C) For the equation (x^2 - 4x - 5) / (x - 5) = x + 1, we simplify the left side by factoring the numerator, (x-5)(x+1), and canceling out (x-5) from the numerator and denominator. This leaves us with x+1 = x+1, which is true for all values of x.

D) For the equation (3a^2 - ac + 2c - 6a) / (3a - c) = a - 2, we simplify the left side by factoring the numerator, (3a-2)(a-c), and canceling out (3a-c) from the numerator and denominator. This leaves us with a-2 = a-2, which is again true for all values of a.