Problem 38 - 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. $$\frac{4}{x}=\frac{9}{5}-\frac{7 x-4}{5 x}$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{4}{x} = \frac{9}{5} - \frac{7x-4}{5x}\) is \(x = \frac{2}{9}\) as long as \(x \neq 0\).

Step 1: Identify Variables in Denominators

The denominators of the fractions in the equation have the variables \(x\) and \(5x\). We find the values of \(x\) that make these denominators zero. These values are the restrictions on the variable. For \(x\), the value that makes it zero is \(x = 0\). As for \(5x\), the restriction is still \(x = 0\) because dividing by zero is undefined.

Step 2: Clear the Denominator

To get rid of the denominators, we find a common denominator for all the fractions in the equation. Both \(x\) and \(5x\) can be divided by \(5x\), so we multiply both sides of the equation by \(5x\): \[5x * \frac{4}{x} = 5x * (\frac{9}{5}-\frac{7 x-4}{5 x}) \] This simplifies to: \[20x = 9x - (7x - 4)\]

Step 3: Simplify and Solve for Variable

Now we simplify and collect like terms: \[20x = 9x - 7x + 4 \] \[20x = 2x + 4 \] Now we can solve for \(x\): \[18x = 4 \] \[x = \frac{4}{18} = \frac{2}{9}\]

Step 4: Check for Restrictions

Finally, we verify our solution against the restrictions discovered in the first step. As \(x = \frac{2}{9}\) does not equal to \(0\), the solution fits the restrictions.

Key Concepts

Variables in Denominators

Rational equations are equations that involve fractions whose numerators or denominators contain algebraic expressions or variables. In these equations, variables can often be found in the denominators. Identifying these variables is crucial for solving the equations because there are specific restrictions to consider—specifically, the values that would make the denominator zero.

Imagine the simple fraction \(\frac{4}{x}\). If the variable \(x\) in the denominator becomes zero, this makes the mathematical expression undefined because division by zero is not possible. Thus, for the equation \(\frac{4}{x} = \frac{9}{5} - \frac{7x-4}{5x}\), we find that both denominators \(x\) and \(5x\) will become zero when \(x = 0\). This makes \(x = 0\) a restriction for this problem. Ensuring that the variable in the denominator never reaches these restricted values is a key step in working with rational equations.

Common Denominator

When working with rational equations, finding a common denominator can be highly beneficial. A common denominator allows us to eliminate the fractions by multiplying through it, simplifying the equation into a form that is easier to handle.

In our problem, the denominators are \(x\) and \(5x\). To clear these denominators, we look for a common expression that encompasses both denominators' terms. In this case, \(5x\) serves as a common denominator. By multiplying every term in the equation by \(5x\), we effectively eliminate the fractions:

• \(5x \cdot \frac{4}{x} = 20\)
• \(5x \cdot \frac{9}{5} = 9x\)
• \(5x \cdot \frac{7x-4}{5x} = 7x - 4\)

These calculations transform the equation into a faction-free version, making the simplification and solution process much more straightforward.

Restrictions on Variables

Understanding restrictions on variables is crucial to solving rational equations correctly. Restrictions arise from the variables in the denominators being equal to zero, causing the equation to become undefined. To avoid these pitfalls, we identify these problematic values early in the solving process.

In our given equation, the restrictions are determined by setting each denominator to zero. For \(x\), this results in \(x = 0\). For \(5x\), the zero occurs again at \(x = 0\), reinforcing the restriction. After solving the equation, it is essential to check that the solution does not violate these restrictions. Failure to do so might lead to an invalid solution.

Solving Algebraic Equations

Once fractions in a rational equation are cleared, the resulting algebraic equation must be simplified and solved. This involves combining like terms and isolating the variable on one side of the equation.

Starting with the transformed equation from our example, we have:\[20x = 9x - (7x - 4)\]Simplifying the right side gives us \(20x = 9x - 7x + 4\), which we further reduce to:\[20x = 2x + 4\]To solve for \(x\), we isolate it by subtracting \(2x\) from both sides, resulting in \(18x = 4\). Dividing through by 18 gives us:\[x = \frac{4}{18} = \frac{2}{9}\]It is vital to check if this solution aligns with our earlier restrictions. Since \(x = \frac{2}{9}\) is not zero, it meets the restrictions, confirming it as a valid solution.