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Chapter 6: Problem 69

Solve each proportion and check. \(\frac{x}{7}=\frac{x+14}{5}\)

Short Answer

Expert verified
The value of x which satisfies the given proportion is -49.

Step by step solution

01

Cross Multiply

Cross multiplication is a technique used in solving such proportions. In this case, it would mean multiplying \(x\) by \(5\) and \(7\) by \(x+14\) to yield the equation \(5x=7(x+14)\).
02

Distribute the 7

Apply the distributive property on \(7(x+14)\) to simplify the right side of the equation. This gives us \(5x=7x+98\).
03

Isolate x

To isolate \(x\), subtract \(7x\) from both sides of the equation. This gives us \(-2x=98\).
04

Solve for x

Divide both sides of the equation \(-2x=98\) by -2, to get \(x=-49\).
05

Check the solution

To confirm if the obtained solution \(x=-49\) is correct, substitute \(x\) by \(-49\) in the original proportion \(\frac{x}{7}=\frac{x+14}{5}\), to get \(\frac{-49}{7}=\frac{-49+14}{5}\). Simplify both sides, and if both sides have an equal value, then the answer is correct.

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Key Concepts

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

Cross Multiplication
Cross multiplication is a powerful technique used to solve proportions easily and efficiently. It involves rearranging the terms in a proportion, which is essentially an equation stating that two ratios are equivalent.
For example, in the given exercise, the proportion is \(\frac{x}{7} = \frac{x+14}{5}\). Cross multiplication helps eliminate fractions by multiplying both sides of the equation by the product of the denominators.
  • First, multiply the numerator of the first fraction by the denominator of the second fraction: \(x \times 5\).
  • Next, multiply the numerator of the second fraction by the denominator of the first fraction: \((x + 14) \times 7\).
This results in an equation where both sides mirror each other: \(5x = 7(x + 14)\). Now, the equation no longer has fractions, making it easier to solve.
Distributive Property
The distributive property is a useful algebraic property that helps us simplify expressions and solve equations. In the context of this exercise, we apply the distributive property to \(7(x + 14)\).
Here's how it works:
  • Multiply the outside number, which is \(7\), by each term inside the parentheses: \(x\) and \(14\).
  • This gives us \(7 \times x + 7 \times 14\).
  • Simplify the expression to get \(7x + 98\).
Applying the distributive property is critical as it allows us to break down the expression into smaller, more manageable pieces. Once simplified, we obtain \(5x = 7x + 98\), making it easier to proceed with solving the equation.
Isolate Variable
Isolating the variable is a vital step in solving equations, especially when you want to find the value of the variable. In this exercise, we need to isolate \(x\) in the equation\(5x = 7x + 98\).
To isolate \(x\), we need to get all terms containing \(x\) on one side:
  • Subtract \(7x\) from both sides to start: \(5x - 7x = 98\).
  • This simplifies to \(-2x = 98\).
To finish isolating \(x\), divide both sides by \(-2\):
  • \(-2x \div -2 = 98 \div -2\).
  • Solving this gives us \(x = -49\).
By isolating the variable, we can solve for \(x\) and find the specific value that satisfies the original proportion.
Check Solution
Checking the solution is an important step to ensure the value found is correct. In this case, we need to verify that \(x = -49\) satisfies the original proportion\(\frac{x}{7} = \frac{x+14}{5}\).
Substitute \(x\) with \(-49\) back into the original proportions:
  • The left side becomes: \(\frac{-49}{7}\).
  • The right side becomes: \(\frac{-49+14}{5}\).
Simplify both sides:
  • \(\frac{-49}{7} = -7\).
  • \(\frac{-35}{5} = -7\).
As both sides equal \(-7\), the solution is verified to be correct. Checking the solution reassures accuracy and confirms that our calculations were performed properly.

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Most popular questions from this chapter

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}-3 x=18\)

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I solved \(-2 x+5 \geq 13\) and concluded that \(-4\) is the greatest integer in the solution set.

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