Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$8-\frac{5}{x}=2+\frac{3}{x}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{3}\).

Step 1: Isolate the terms with variable

Start by moving all terms involving the variable \(x\) to one side of the equation. Subtract \(\frac{3}{x}\) from both sides:\[8 - \frac{5}{x} - \frac{3}{x} = 2\] Simplify the left side:\[8 - \frac{8}{x} = 2\]

Step 2: Isolate the variable term

In this step, move the constant term to the other side by subtracting 8 from both sides of the equation:\[-\frac{8}{x} = 2 - 8\] Simplify the right side:\[-\frac{8}{x} = -6\]

Step 3: Solve for the variable

Next, we want to solve for \(x\) by eliminating the fraction. Multiply both sides of the equation by \(-x\):\[8 = 6x\] Now, solve for \(x\) by dividing both sides by 6:\[x = \frac{8}{6}\] Simplify the fraction:\[x = \frac{4}{3}\]

Step 4: Check the solution

Substitute \(x = \frac{4}{3}\) back into the original equation to ensure it's correct:\[8 - \frac{5}{\frac{4}{3}} = 2 + \frac{3}{\frac{4}{3}}\]Calculate both sides:\[8 - \frac{15}{4} = 2 + \frac{9}{4}\]Convert the whole numbers to fractions with a denominator of 4:\[\frac{32}{4} - \frac{15}{4} = \frac{8}{4} + \frac{9}{4}\]Simplifying both sides gives:\[\frac{17}{4} = \frac{17}{4}\]Both sides are equal, confirming that \(x = \frac{4}{3}\) is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra, providing a way to find unknown values in mathematical expressions. An equation is a statement that asserts the equality of two expressions. To solve an equation, we need to find the value of the variable that makes the equation true. Equations can take many forms, from simple linear equations to more complex quadratic or rational equations. The primary goal is to isolate the variable on one side of the equation. This is usually done through a series of operations like addition, subtraction, multiplication, and division.

• Identify the type of equation you are dealing with.
• Apply inverse operations to move terms, aiming to have the variable alone on one side.
• Ensure all terms are simplified correctly to avoid errors.

Solving equations can sometimes be straightforward but can also involve multiple steps, especially when dealing with fractions or more complex expressions.

Rational Equations

Rational equations involve at least one fraction where the numerator or the denominator contains a variable. Solving rational equations requires paying attention to the denominators and understanding how they affect the overall equation. Start by identifying the fractions in the equation and consider strategies to eliminate them. This often involves finding a common denominator or multiplying through by a term that will clear the fractions.

• Review basic operations with fractions, including finding a common denominator.
• Be cautious of values for which the denominator becomes zero, as these are not valid solutions.
• Rewriting the equation in a non-fractional form can simplify the solving process.

Rational equations can sometimes have restrictions on the variable, meaning certain values are not allowed because they would make the denominator zero. Always check for these restrictions by setting the denominators to zero and solving.

Variable Isolation

To solve any equation, isolating the variable is a critical step. In rational equations, this step might involve combining terms and removing fractional parts. When isolating the variable:

• Move all terms involving the variable to one side of the equation.
• Perform any operations needed to get the variable term alone, such as adding or subtracting terms, and multiplying or dividing.
• In the case of fractions, eliminate the fractional form by appropriate multiplications.

Variable isolation simplifies the equation to a basic form where solving for the variable becomes straightforward. This may involve combining like terms or simplifying expressions until the variable can be directly solved for.

Checking Solutions

Once you find a solution to an equation, it is crucial to verify it. Checking your solution ensures that no mistakes were made during the solving process and that the solution satisfies the original equation. For checking solutions:

• Substitute the found value back into the original equation.
• Simplify both sides of the equation to see if they equal.
• If they are not equal, recheck your solving steps for any errors.

Remember, especially with rational equations, to check if your solution introduces any invalid states, like a zero in the denominator. Confirming your solution guarantees accuracy and reinforces understanding of the process.