Problem 32 - Algebra and Trigonometry

Question

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{5}{x}=\frac{10}{3 x}+4$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{12}\). The restriction for the variable \(x\) in this exercise is \(x ≠ 0\).

Step 1: Identify the Restrictions on the Variable

To find the restrictions, set the denominators equal to zero and solve for \(x\). Here, the denominators are \(x\) and \(3x\) respectively. If \(x=0\), both denominators would be zero which is not allowed in mathematics. Hence, \(x ≠ 0\) is a restriction of this problem.

Step 2: Clear the Denominator

Multiply each term on both sides by 3x to eliminate the denominator, this will leave only integers and the variable \(x\). This gives us: 15 = 10 + 12x.

Step 3: Simplify and Solve for the Variable

Reformulate the equation from step 2 into 12x = 15 - 10 which simplifies to 12x = 5. Divide both sides by 12, we get \(x = \frac{5}{12}\)

Step 4: Check If the Solution Is Valid

One has to ascertain that the located answer does not violate the restriction. In this case, \(x = \frac{5}{12}\) does not violate the restriction \(x ≠ 0\), hence, it is a valid solution.

Key Concepts

Restrictions on Variables

When solving rational equations, a crucial early step is identifying any restrictions on variables. These restrictions are values that would make any denominators in the equation equal to zero. As denominators can't be zero in any valid mathematical expression—because division by zero is undefined—we must exclude these values from the solution set.

For instance, in the exercise \( \frac{5}{x} = \frac{10}{3x} + 4 \), the denominators are \( x \) and \( 3x \) which immediately indicate that \( x \) cannot be zero. Mathematically, we express this as \( x eq 0 \). This forms the basis of our problem-solving approach, ensuring that the solutions we find are mathematically valid.

Clearing Denominators

An efficient strategy to solve rational equations is clearing denominators. This process involves multiplying every term in the equation by the least common denominator (LCD) to transform the rational equation into a simpler algebraic equation without fractions.

In our given exercise, the LCD is \( 3x \) because it is the smallest number that both \( x \) and \( 3x \) can divide into. By multiplying each term by \( 3x \) we eliminate the fractions, facilitating a straightforward solution process. This step is crucial as it paves the way to deal with simpler algebraic forms instead of more complex rational ones.

Simplify Algebraic Expressions

After clearing the denominators, the equation frequently needs to be simplified. Simplifying algebraic expressions involves combining like terms and reducing equations to their most basic forms. The goal is to isolate the variable on one side of the equation to solve for it.

In our exercise, after multiplying the terms by \( 3x \) and rearranging, we simplify the equation to \( 12x = 5 \), a much clearer and simpler form. We then continue the simplification process by dividing both sides by 12 to solve for \( x \). Simplification is vital for maintaining clarity and accuracy in solving algebraic problems.

Validate Algebraic Solutions

The final step in solving an algebraic problem is to validate the solution. Every solution must be checked against the original problem's restrictions to make sure it does not make any denominators zero. This confirmation ensures that the solution is applicable and does not lead to an undefined mathematical expression.

In the context of our example, we've determined that \( x = \frac{5}{12} \). It's essential to verify this solution does not violate the restriction \( x eq 0 \). Since \( \frac{5}{12} \) is not zero, it meets the criteria, and thus we can confidently state that \( x = \frac{5}{12} \) is a valid and legitimate solution to the rational equation.