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Chapter 7: Problem 22

Solve. If no solution exists, state this. $$\frac{x}{2}=\frac{18}{x}$$

Short Answer

Expert verified
The solutions are \(x = 6\) and \(x = -6\).

Step by step solution

01

Cross Multiply

Multiply both sides of the equation by the denominators to eliminate the fractions. \[\frac{x}{2} = \frac{18}{x} \rightarrow x \times x = 18 \times 2\]This simplifies to \[x^2 = 36\]
02

Solve for x

To solve for \(x\), take the square root of both sides of the equation. \[\begin{aligned} x^2 &= 36 \ \rightarrow x &= \text{卤}\text{ }\text{鈭殅36 \ \rightarrow x &= \text{卤}\text{ }6 \end{aligned}\]
03

Verify Solutions

Substitute both values back into the original equation to ensure they satisfy the equation. Check \(x = 6\):\[\frac{6}{2} = \frac{18}{6} \rightarrow 3 = 3\]This is true. Check \(x = -6\): \[\frac{-6}{2} = \frac{18}{-6} \rightarrow -3 = -3\]This is true.
04

State the Final Solutions

Both \(x = 6\) and \(x = -6\) are solutions to the original equation.

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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 method used to solve equations involving fractions. It helps to eliminate the fractions, making the equation easier to solve. In this exercise, we start with the equation \(\frac{x}{2} = \frac{18}{x}\). To cross multiply, you multiply the numerator of each fraction by the denominator of the other fraction:
  • Multiply \(x\) by \(x\)
  • Multiply 18 by 2
This gives us the equation: \(x \times x = 18 \times 2\)
Simplifying further, this becomes: \(x^2 = 36\). Now, we have rid the equation of fractions, making it a polynomial equation. Cross multiplication is powerful because it converts complex fraction problems into simpler forms.
Square Roots
Square roots are instrumental in solving quadratic equations like \(x^2 = 36\). To isolate \(x\), take the square root of both sides of the equation. Applying the square root gives us:
\(x = \pm \sqrt{36}\)
The square root of 36 is 6, but don't forget the 卤 symbol. It indicates there are two possible solutions:
  • \(x = 6\)
  • \(x = -6\)
Square roots help us find both positive and negative values for \(x\) that satisfy the equation. Taking the square root is a critical step in solving quadratic equations after simplifying them by cross-multiplying.
Verifying Solutions
After finding possible solutions, it's important to verify them by substituting them back into the original equation. This ensures the solutions are correct. Let's verify both \(x = 6\) and \(x = -6\):
  • For \(x = 6\):
    Substitute into the original equation: \(\frac{6}{2} = \frac{18}{6}\)
    Simplify: \(3 = 3\)
    This is true.
  • For \(x = -6\):
    Substitute into the original equation: \(\frac{-6}{2} = \frac{18}{-6}\)
    Simplify: \(-3 = -3\)
    This is true.
Both solutions satisfy the original equation, so they are valid. Verifying solutions prevents errors and confirms that the computed values solve the initial problem correctly.

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Most popular questions from this chapter

Find an equation of variation in which: \(y\) varies jointly as \(w\) and the square of \(x\) and inversely as \(z,\) and \(y=49\) when \(w=3, x=7\) and \(z=12\)

The height of a triangle is \(3 \mathrm{cm}\) longer than its base. If the area of the triangle is \(54 \mathrm{cm}^{2},\) find the measurements of the base and the height.

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For each pair of functions \(f\) and \(g,\) find all values of a for which \(f(a)=g(a)\). $$f(x)=\frac{12}{x^{2}-6 x+9}$$, $$g(x)=\frac{4}{x-3}+\frac{2 x}{x-3}$$

Find the LCM. $$80,96,108$$
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