Problem 67 - Algebra and Trigonometry

Question

Solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. $$\frac{2 x}{x-3}=\frac{6}{x-3}+4$$

Step-by-Step Solution

Answer

The given equation is an inconsistent equation with no solution since 'x = 3' is excluded.

Step 1: Simplify the Equation

The first step to solving the equation is to simplify it. We can begin by subtracting \(\frac{6}{x-3}\) from both sides to consolidate the terms. This yields: \(\frac{2x - 6}{x - 3} = 4\).

Step 2: Solve for 'x'

Now, let's isolate the variable 'x' by cross multiplying. This gives us: \(2x - 6 = 4(x - 3)\). Then, expand the right side: \(2x - 6 = 4x - 12\). Subtracting 2x from both sides, we get: \(-6 = 2x - 12\). Adding 12 to both sides, we then solve for 'x' to get: x = 3.

Step 3: Check the Equation Type

Since 'x' is in the denominator of the original equation, a valid solution can't make denominator equal to zero. So, 'x = 3' must be excluded, thus there is no solution for this equation, which defines it as an inconsistent equation.

Key Concepts

Identity Equation

An identity equation is a type of equation that is true for all values of the variable. Think of it as a statement that is always true, no matter what number you plug in for the variable. For example, the equation \( 2x + 3 = 2x + 3 \) is an identity equation. Here, no matter what value 'x' is, both sides will always be equal.

• Identity equations are helpful because they show forms that are always reliable.
• These equations usually result in something like \( 0 = 0 \) when simplified fully.

It’s important to note that the equation \( \frac{2x}{x-3} = \frac{6}{x-3} + 4 \) is not an identity. This is because it does not hold true for all values of 'x'. Certain restrictions exist, such as 'x' not equalling 3, which can make the equation incorrect. Thus, this type of equation cannot be categorized as an identity equation.

Conditional Equation

A conditional equation is true for specific values of the variable. It differs from an identity equation in that not all values of the variable satisfy it. Taking the exercise example, a conditional equation would have offered a valid solution that does not violate the conditions of the equation.

• Conditional equations work under specific circumstances or conditions.
• For example, solving \( x + 2 = 5 \) results in \( x = 3 \), a single specific value.

In our problem, the equation \( \frac{2x}{x-3} = \frac{6}{x-3} + 4 \) was expected to be conditional upon solving, except 'x = 3' made the equation undefined because substituting 3 causes the denominator to be zero. Hence, this one has no valid solution under its conditions.

Rational Equations

Rational equations are equations containing fractions whose numerators and/or denominators are polynomials. The core aspect with rational equations is to be careful with values that make any denominator zero, as these are not allowed.

• To solve rational equations, clear the fractions by multiplying through by the least common denominator.
• Always check solutions to ensure they do not make any term in the denominator equal to zero.

In the exercise provided, \( \frac{2x}{x-3} = \frac{6}{x-3} + 4 \) is a rational equation. Notice that not all solutions are valid because of the variable 'x' being in the denominator. Particularly, 'x = 3' leads to division by zero in the original equation, which is not permitted. This issue results in this particular problem having no valid solutions, classifying it as inconsistent.