Problem 37 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(\frac{2}{x}-5=\frac{6}{x}+4\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{4}{9}\).

Step 1: Identify the Linear Form

The given equation is \(\frac{2}{x} - 5 = \frac{6}{x} + 4\). We see that the terms \(\frac{2}{x}\) and \(\frac{6}{x}\) need to be grouped together to identify its linear form. This can be done by keeping all terms involving \(x\) on one side.

Step 2: Simplify the Equation

Subtract \(\frac{6}{x}\) from both sides of the equation to isolate the terms involving \(x\): \[ \frac{2}{x} - \frac{6}{x} = 4 + 5 \] Simplify the equation by combining similar terms: \[ \frac{2}{x} - \frac{6}{x} = 9 \]

Step 3: Combine the Fractions

Combine the fractions on the left side: \[ \frac{2-6}{x} = 9 \] which simplifies to: \[ \frac{-4}{x} = 9 \]

Step 4: Solve for \(x\)

To solve for \(x\), multiply both sides by \(-x\) to get rid of the fraction: \[ -4 = 9x \]

Step 5: Isolate \(x\)

Divide both sides by 9 to solve for \(x\): \[ x = \frac{-4}{9} \]

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental skill that simplifies expressions and equations. When we combine like terms, we are essentially grouping similar items together to make the equation simpler and more manageable. In our exercise, we looked at terms \( \frac{2}{x} \) and \( \frac{6}{x} \). Both terms are fractional and share the same denominator.

To combine them, subtract the coefficients while keeping the common denominator intact. So, in this case:

• Subtract \( 6 \) from \( 2 \) to get \( -4 \).
• The expression becomes \( \frac{2}{x} - \frac{6}{x} = \frac{-4}{x} \).

Through this process, we were able to group the fractions into a simpler form that is easier to solve. Remember that combining like terms effectively reduces the complexity of equations and is crucial for moving forward in solving them.

Fractional Equations

Fractional equations can be a bit intimidating at first, as they involve fractions which many find tricky. However, understanding and manipulating these equations can be straightforward with the right steps.
In our exercise, the equation given is a fractional one:

• Original equation: \( \frac{2}{x} - 5 = \frac{6}{x} + 4 \)

The key to handling fractional equations is to clear the fractions as early as possible. You can achieve this by finding a common denominator or multiplying through by a term that cancels out the denominator. In this situation, after we combine like terms, we were left with:

• \( \frac{-4}{x} = 9 \)

To clear the fraction, multiply both sides by the variable present in the denominator \( -x \) which simplifies the equation greatly, converting it into a linear form that's easier to solve:

Isolating Variables

Isolating variables is a key concept when solving equations. The goal is to transform the equation so that the variable is by itself on one side of the equals sign. This allows you to identify the solution directly.

In the exercise, after simplifying the fractional equation, we reached the stage where we had:

• \( -4 = 9x \)

To isolate \( x \), we continued solving by dividing both sides of the equation by the coefficient in front of \( x \). Here, it's 9, making sure we perform the operation on both sides to maintain equality:

• \( x = \frac{-4}{9} \)

Isolating the variable is crucial because it provides the solution. Always remember to check your solution by plugging it back into the original equation to verify accuracy.