Recognize Identities, Conditional Equations, and Inconsistent Equations
Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 7 } { y + 3 } - \frac { 2 } { y - 3 } = \frac { 3 } { y ^ { 2 } - 9 }$$
A)Identity
B)Conditional equation
C)Inconsistent equation

Question

Quizplus · Accepted Answer

To determine whether the equation is an identity, a conditional equation, or an inconsistent equation, we can start by simplifying it.

First, we need to find a common denominator for the two fractions on the left side of the equation:

$$\frac { 7 } { y + 3 } - \frac { 2 } { y - 3 } = \frac { 3 } { y ^ { 2 } - 9 }$$

LCD: (y+3)(y-3)

$$\frac { 7(y-3) } { (y + 3)(y-3) } - \frac { 2(y+3) } { (y + 3)(y-3) } = \frac { 3 } { (y + 3)(y-3) }$$

Simplify:

$$\frac { 7y - 21 - 2y - 6 } { (y + 3)(y-3) } = \frac { 3 } { (y + 3)(y-3) }$$

$$\frac { 5y - 27 } { (y + 3)(y-3) } = \frac { 3 } { (y + 3)(y-3) }$$

Now, we can see that the denominators on both sides of the equation are the same, so we can compare the numerators. If the numerators are equal, then the equation is an identity. If the numerators are not equal but there is still some value of y that makes both sides of the equation equal, then the equation is a conditional equation. If there is no such value of y, then the equation is inconsistent.

In this case, we can see that the numerators are not equal, so it is not an identity. However, we can see that we can multiply both sides by the common denominator to eliminate the fractions:

$$5y - 27 = 3$$

$$5y = 30$$

$$y = 6$$

Since there is a specific value of y that makes both sides of the equation equal (y=6), the equation is a conditional equation. Therefore, the answer is choice B.

Recognize Identities, Conditional Equations, and Inconsistent Equations Determine Whether the Equation

Recognize Identities, Conditional Equations, and Inconsistent Equations
Determine Whether the Equation