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Chapter 10: Problem 7

Solve for \(x\) in each of the following proportions. $$\frac{1}{x}=\frac{3}{2 x-1}$$

Short Answer

Expert verified
The solution is \(x = -1\).

Step by step solution

01

Set Up the Equation

Take the given proportion \(\frac{1}{x} = \frac{3}{2x-1}\). Our goal is to solve for \(x\).
02

Cross Multiply

Cross multiply to eliminate the fractions: \(1 \cdot (2x - 1) = 3 \cdot x\). This simplifies to the equation: \(2x - 1 = 3x\).
03

Simplify the Equation

Subtract \(2x\) from both sides to isolate the terms with \(x\) on one side of the equation: \( -1 = 3x - 2x\). This simplifies to \(-1 = x\).
04

Final Check

Substitute \(x = -1\) back into the initial fraction \(\frac{1}{x} = \frac{3}{2x-1}\) to ensure both sides are equal. That gives \(-1 = \frac{3}{-3}\), verifying both sides are equal, confirming the solution.

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Key Concepts

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

Proportions
Proportions are a powerful tool in algebra, allowing us to solve problems involving ratios. A proportion is an equation in which two ratios are set equal to each other. In a more straightforward sense, it suggests how two quantities relate to each other in the same manner. This concept is essential because it helps us find unknown values, like the variable in our exercise.
  • For example, if we know two parts of a proportion, we can find the missing part using algebraic manipulation.
  • In our exercise, the given proportion is \(\frac{1}{x} = \frac{3}{2x-1}\).
When setting up proportions, always ensure that the relationship between the quantities is set up correctly. This ensures our calculations will lead to the correct solution.
Cross Multiplication
Cross multiplication is a technique used to solve proportions efficiently by eliminating fractions. It involves multiplying across the equal sign diagonally. This technique helps to simplify the expression into an algebraic equation that is easier to solve.
  • In the exercise, we apply cross multiplication to the equation \(\frac{1}{x} = \frac{3}{2x-1}\).
  • We cross multiply to get \(1 \cdot (2x - 1) = 3 \cdot x\).
This step transforms the equation from a proportion with fractions into a more accessible linear equation, \(2x - 1 = 3x\). By doing so, the equation becomes much easier to manage, setting the stage for simplification.
Equation Simplification
Once we have an algebraic equation, the next step is to simplify it to solve for our variable, \(x\). Simplification involves performing algebraic operations to isolate the variable on one side of the equation.
  • With our equation \(2x - 1 = 3x\), we subtract \(2x\) from both sides to simplify it.
  • This manipulation leaves us with \(-1 = x\).
The process of simplification allows us to clearly see that \(x = -1\) is the solution. It is crucial to always perform a final check by substituting the solution back into the original equation to verify accuracy. By substituting \(x = -1\), both sides equal \(-1\), confirming the correctness of our solution.

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Most popular questions from this chapter

Refer to the given correspondences to complete the following statements. If \(\triangle B O G \sim \triangle A R T\), then \(\frac{\mathrm{OG}}{\mathrm{RT}}=\frac{\mathrm{BG}}{\mathrm{?}}\)

Tell what can be done to both sides of the equation $$ a d=b c $$ to get each of the following equations. $$\text { Example: } \frac{a}{b}=\frac{c}{d}$$ $$ \begin{array}\text{Answer: Divide} \text{ by } { bd } \\ \frac{a d}{b d}=\frac{b c}{b d}, \text { and so } \frac{a}{b}=\frac{c}{d} \end{array} $$ $$\frac{d}{c}=\frac{b}{a}$$

Two ratios that are famous in the history of the number \(\pi\) are \(\frac{22}{7}\) and \(\frac{355}{113}\) Express \(\frac{355}{113}\) in decimal form to the nearest hundredth.

Solve for \(x\) in each of the following proportions. $$\frac{6}{17}=\frac{2 x}{51}$$

Example: 12 and 15 $$ \begin{aligned} \text { Solution: } \frac{12}{x} &=\frac{x}{15} \\ x^{2} &=180 \\ x &=\sqrt{180} \text { (because } x \text { is positive }) \\ &=\sqrt{36} \cdot 5 \\ &=6 \sqrt{5} \end{aligned} $$ 3 and 27
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