Problem 43 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

a = 6 or a = -6

Step 1: 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: \(a \times a = 9 \times 4\).

Step 2: Simplify

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

Step 3: Solve for a

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

Key Concepts

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 \({a = \text{±}\begin{sqrt}{36}}\). 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.