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Chapter 6: Problem 43

Solve each equation. $$\frac{a}{9}=\frac{4}{a}$$

Short Answer

Expert verified
a = 6 or a = -6

Step by step solution

01

Cross Multiply

To solve the equation \(\frac{a}{9} = \frac{4}{a}\), start by cross-multiplying to eliminate the fractions. Multiply both sides by 9 and a: \(a \times a = 9 \times 4\).
02

Simplify

Simplify the equation from the cross-multiplication: \(a^2 = 36\).
03

Solve for a

Next, solve for \a\ by taking the square root of both sides of the equation: \(\text{If} \ a^2 = 36, \ \text{then} \ a = \ \text{卤}\begin{sqrt}{36}\).So, \ a = 6 \, or \ a = -6 \.

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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 eliminate the denominators, making the equation easier to work with. In the exercise \(\frac{a}{9} = \frac{4}{a}\), we can use cross-multiplication to get rid of the fractions:
  • Step 1: Identify the fractions on both sides of the equation.
  • Step 2: Multiply both sides by the denominators. Here, multiply by 9 and a.
  • Step 3: The equation becomes \({a \times a = 9 \times 4}\).
This process results in a simpler equation without fractions. Cross-multiplying makes it straightforward to proceed to the next steps.
Solving Quadratic Equations
Once the equation from the cross-multiplication step is simplified to \({a^2 = 36}\), we recognize it as a quadratic equation. We solve quadratic equations by finding the values of the variable that satisfy the equation.
  • Step 1: Identify the quadratic equation format. Here, it is \({a^2 = 36}\)
  • Step 2: Solve for the variable a. Take the square root of both sides.
  • Step 3: Considering both positive and negative roots, we get \({a = \text{卤}\begin{sqrt}{36}}\). This gives us \({a = 6}\) or \({a = -6}\).
Make sure to always consider both the positive and negative solutions resulting from taking the square root of both sides of the equation.
Simplifying Equations
Simplifying equations is a crucial step that helps in reducing complex expressions to simpler forms, making the solving process manageable.
  • Step 1: After cross-multiplying, simplify the right-hand side. Here, \({9 \times 4 = 36}\).
  • Step 2: Identify and simplify common terms or factors. In our example, the equation \({a^2 = 36}\) already appears in its simplest quadratic form.
  • Step 3: Apply algebraic rules accurately, such as factoring, to get the final simplified results.
Simplifying equations systematically reduces opportunities for mistakes and leads to finding solutions effectively.

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Most popular questions from this chapter

Perform the indicated operations. $$\frac{\left(a^{2} b^{3} c\right)^{2}}{\left(-2 a b^{2} c\right)^{3}} \cdot \frac{\left(a^{3} b^{2} c\right)^{3}}{(a b c)^{4}}$$

Find the solution set to each equation. $$\frac{b}{5}=\frac{-3}{7}$$

Solve each equation. $$\frac{x-2}{x-6}-\frac{4}{x}=\frac{24}{x^{2}-6 x}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{5 x-5 y}{x} \cdot \frac{1}{x-y}$$

Solve each equation. $$\frac{-3}{2 x}=\frac{1}{-5}$$
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