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Chapter 6: Problem 6

Solve each equation. $$ \frac{5}{3 x}+1=\frac{7}{6} $$

Short Answer

Expert verified
The solution to the equation is x = 10.

Step by step solution

01

Eliminate Fractions

First, let's eliminate the fractions by finding a common denominator for all terms. The common denominator for 3x and 6 is 6x. Multiply every term in the equation by 6x:\[6x \left( \frac{5}{3x} \right) + 6x(1) = 6x \left( \frac{7}{6} \right).\]This simplifies to:\[10 + 6x = 7x.\]
02

Isolate the Variable

Next, we need to solve for x. Subtract 6x from both sides:\[10 = 7x - 6x.\]Simplify to:\[10 = x.\]
03

Verify the Solution

Substitute x = 10 back into the original equation to verify the solution:\[\frac{5}{3 \times 10} + 1 = \frac{7}{6}.\]Simplify the left side:\[\frac{5}{30} + 1 = \frac{1}{6} + 1 = \frac{1}{6} + \frac{6}{6} = \frac{7}{6}.\]Since both sides are equal, x = 10 is the correct solution.

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Key Concepts

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

Fraction Elimination
When solving equations with fractions, the first step is often to eliminate the fractions. This process makes the equation easier to handle and prevents potential mistakes during calculations. To eliminate fractions, identify a common multiple for the denominators of the fractions involved. In this exercise, we have the fractions \( \frac{5}{3x} \) and \( \frac{7}{6} \). The denominators here are \( 3x \) and \( 6 \), respectively.

- The common denominator for these terms is \( 6x \).
- Multiply every term in the equation by this common denominator to clear the fractions.

So, multiplying each term by \( 6x \) transforms the original equation. This step results in a simpler equation without fractions, making further calculations straightforward.
Common Denominator
Understanding common denominators is a crucial step in dealing with equations involving fractions. A common denominator is a shared multiple of the denominators of the fractions within an equation. To find a common denominator, evaluate the denominators and identify the lowest common multiple (LCM).

- Look at each denominator individually; in this case, \( 3x \) and \( 6 \).
- The common denominator needs to be a multiple of both, which is why \( 6x \) was chosen.

Once you've found a common denominator, multiply each term of the equation by this number. This action converts the equation into one without fractions, significantly simplifying it down into a more manageable form. This technique is valuable for almost any scenario involving fractions in equations.
Verifying Solutions
After finding a potential solution to an equation, verifying the solution is essential. This process ensures that the solution is correct and that no calculation mistakes were made. Verification involves substituting the found value back into the original equation to check for consistency.

For this particular equation, substituting \( x = 10 \) back into the original equation, \( \frac{5}{3x} + 1 = \frac{7}{6} \), helps in confirming that both sides of the equation are equal.

- Plug \( x = 10 \) into the equation.
- Simplify the fractions on both sides.

If both sides of the equation are equal after simplifying, the solution is verified as correct. This step provides confidence in the solution and affirms the process used to arrive at it.

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Most popular questions from this chapter

eBay is the leading online auction house. eBay's annual net profit can be modeled by the polynomial function \(P(x)=0.48 x^{3}+2.06 x^{2}+141 x+9.71,\) where \(P(x)\) is net profit in millions of dollars and \(x\) is the number of years since 2000 . eBay's annual revenue can be modeled by the function \(R(x)=1011 x-288,\) where \(R(x)\) is revenue in millions of dollars and \(x\) is years after 2000 . (Source: eBay, Inc., annual reports \(2000-2010\) ) a. Given that Net profit margin \(=\frac{\text { net profit }}{\text { revenue }}\) write a function, \(m(x),\) that models eBay's net profit margin. b. Use part (a) to predict eBay's profit margin in 2015 . Round to the nearest hundredth.

Divide. $$ \left(5 x^{4}-2 x^{2}+10 x^{3}-4 x\right) \div(5 x+10) $$

Explain an advantage of using synthetic division instead of long division.

Find each square root. See Section 1.3. $$ \sqrt{\frac{4}{9}} $$

A board of length \(\left(3 x^{4}+6 x^{2}-18\right)\) meters is to be cut into three pieces of the same length. Find the length of each piece.
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