Problem 46 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{y}{4}=\frac{4}{y} $$

Step-by-Step Solution

Answer

The values of \( y \) are \( 4 \) and \( -4 \).

Step 1: Cross-Multiply the Proportion

Start by cross-multiplying the terms of the proportion: \( \frac{y}{4} = \frac{4}{y} \). This means multiplying the numerator of one fraction by the denominator of the other and setting the products equal: \( y \cdot y = 4 \cdot 4 \).

Step 2: Simplify the Equation

Perform the multiplications from Step 1 to simplify the equation. You get: \( y^2 = 16 \).

Step 3: Solve for y

Take the square root of both sides of the equation from Step 2 to solve for \( y \). \( y = \pm \sqrt{16} \).

Step 4: Find the Values of y

Calculate the square root from Step 3: \( y = \pm 4 \). This means that \( y \) can be either \( 4 \) or \( -4 \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful tool often used in solving proportions. It is especially useful when you have two fractions set equal to each other, as in the original exercise. Cross-multiplication involves multiplying across the equals sign in a criss-cross manner. This means you multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.

For example, with the proportion \( \frac{y}{4} = \frac{4}{y} \), you multiply \( y \) (the numerator of the first fraction) by \( y \) (the denominator of the second fraction), and \( 4 \) by \( 4 \). This results in the equation \( y \cdot y = 4 \cdot 4 \).

• Cross-multiplication transforms a proportion into a simple equation, making it easier to solve.
• It is crucial to ensure that neither side of the equation equals zero, as dividing by zero is undefined.

Applying this concept helps quickly shift from a complex proportion to a more straightforward polynomial equation.

Solving Equations

Once you have applied cross-multiplication, the next step is to solve the resulting equation. In our case, after cross-multiplying, we obtained \( y^2 = 16 \). Solving equations generally involves isolating the variable, in this case, \( y \).

You simplify this equation by completing the multiplications, as was done in the solution, where \( y \cdot y \) simplifies to \( y^2 \), and \( 4 \cdot 4 \) simplifies to 16.

• The goal is to get the variable on one side of the equation while everything else is on the other side.
• Simplifying an equation is often about performing arithmetic operations like multiplication or division on both sides to maintain equality.

This process sets the stage for determining the possible value(s) of the variable, which will be further refined in subsequent steps.

Square Roots

Taking square roots is a method used to solve equations involving squared terms. Once we have simplified the equation to \( y^2 = 16 \), the challenge is to find \( y \). The square root function helps us step away from the squared variable to find potential solutions for \( y \).

When you take the square root of both sides of an equation, such as here, we perform \( y = \pm \sqrt{16} \). This step is crucial because every positive number has two square roots: one positive and one negative.

• The symbol \( \pm \) indicates that both the positive and negative square roots are valid solutions.
• For \( \sqrt{16} \), the results are 4 and -4, reflecting the two possible values of \( y \).

Understanding how to find square roots allows you to complete equations that include squared variables and reach final solutions easily.