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Chapter 9: Problem 49

Solve the equation for \(\mathrm{x}.\) $$30=\frac{115}{\mathrm{x}}$$

Short Answer

Expert verified
The solution of the equation is \(x=\frac{115}{30}\) or x approximately equals to 3.83 when rounded to two decimal places.

Step by step solution

01

Rewrite the equation to isolate x

To make the equation simpler, multiply both sides of the equation by \(x\), which will remove the denominator on the right side of the equation. The equation will become \(30x = 115\).
02

Solve for x

Now, divide both sides of the equation by 30. This will finally isolate \(x\). You get \(x=\frac{115}{30}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Rational Equations
A rational equation is one that involves fractions with variables in the denominator. In these types of equations, we often need a strategy to cleverly eliminate the fractions so we can solve for the unknown variable. Here鈥檚 a quick breakdown of the approach:
  • Find the Least Common Denominator (LCD): This step is crucial if you have more than one fraction. The LCD of a set of denominators is the smallest expression that all denominators divide into evenly. However, in the given exercise, we only have one fraction, so this step simplifies.

  • Clear the Fraction: You do this by multiplying every term in the equation by the LCD. This will cancel out the denominators. In our exercise, we have one fraction: \( \frac{115}{x} \). To eliminate the denominator, we multiplied both sides by \( x \). This gave us an equation without fractions: \( 30x = 115 \).

  • Solve the Resulting Equation: Once the fraction is cleared, you work with the remaining linear equation to solve for the variable.

By transitioning from a rational equation to a simpler one, solving becomes manageable. Remember that making the equation fraction-free is the first major step to getting closer to our solution.
Isolating Variables
To solve for a specific variable, it needs to be isolated on one side of the equation. The goal is to make the equation state that variable equals something. Here鈥檚 how you can isolate a variable effectively:
  • Identify the Variable to Isolate: In our equation, we need to solve for \( x \).

  • Eliminate Other Terms: Use inverse operations to remove any numbers or other terms that are attached to the variable. This can involve addition, subtraction, multiplication, or division.In the exercise, after clearing the fraction, we had \( 30x = 115 \). To isolate \( x \), we had to divide both sides by 30.

  • Simplify: Perform the operations to fully isolate the variable. Here, dividing \( 115 \) by 30 gave us \( x = \frac{115}{30} \).
Isolating a variable is like peeling away layers one by one until the variable you鈥檙e interested in stands alone. Once it's isolated, you have the solution.
Algebraic Manipulation
This concept involves using mathematical operations to transform an equation into a form that is easier to solve. In algebraic manipulation, you often perform operations like adding, subtracting, multiplying, or dividing terms to simplify the equation. Let's see how it applies:
  • Maintaining Balance: Algebraic manipulation requires maintaining the balance of the equation, meaning whatever you do to one side, you must do to the other.In our example, to eliminate the fraction, we multiplied both sides by \( x \), which is why the equation remained balanced.

  • Using Inverse Operations: Operations are undone using their opposites. For instance, multiplication is undone by division. In our equation, we divided both sides by 30 to undo the multiplication and get \( x \) by itself.

  • Simplifying Expressions: Once the operations are performed, it's crucial to simplify the expression to its easiest form for better clarity.After the division in our example, we ended up with a clear result: \( x = \frac{115}{30} \).
Through algebraic manipulation, equations transform from complex forms to clean, understandable results. This journey involves decisive steps that make the equation easily solvable.

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Most popular questions from this chapter

In Exercises \(3-6,\) determine which of the two acute angles has the given trigonometric ratio. (See Example \(1 .\) ) The cosine of the angle is \(\frac{4}{5}\)

REASONING Explain why it is not possible to ?nd the tangent of a right angle or an obtuse angle.

Prove the Right Triangle Similarity Theorem (Theorem 9.6) by proving three similarity statements. Given \(\triangle\) ABCis a right triangle. Altitude \(\overline{\mathrm{CD}}\) is drawn to hypotenuse \(\overline{\mathrm{AB}}\) . Prove \(\triangle \mathrm{CBD} \sim \triangle \mathrm{ABC} ; \Delta \mathrm{ACD} \sim \triangle \mathrm{ABC}\) \(\Delta \mathrm{CBD} \sim \triangle \mathrm{ACD}\)

The Leaning Tower of Pisa in Italy has a height of 183 feet and is \(4^{\circ}\) off vertical. Find the horizontal distance d that the top of the tower is off vertical.

THOUGHT PROVOKING Simplify each expression. Justify your answer. a. \(\sin ^{-1}(\sin x)\) b. \(\tan \left(\tan ^{-1} y\right)\) c. \(\cos \left(\cos ^{-1} z\right)\)
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