Problem 30 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solutions are \( y = 4 \) and \( y = -4 \).

Step 1: Set up the equation

Given the proportion \( \frac{y}{4} = \frac{4}{y} \), we start by setting up the equation exactly as it is presented.

Step 2: Cross-multiply

To eliminate the fractions, we cross-multiply. This means we multiply \( y \) by \( y \) and 4 by 4, resulting in the equation \( y^2 = 16 \).

Step 3: Solve for y

To find \( y \), we need to solve \( y^2 = 16 \). We do this by taking the square root of both sides, giving us \( y = \pm 4 \).

Step 4: Check solutions

Substitute \( y = 4 \) and \( y = -4 \) back into the original proportion to see if they satisfy the equation. Both values satisfy \( \frac{y}{4} = \frac{4}{y} \), confirming them as solutions.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a powerful technique used in solving proportions. A proportion is an equation where two fractions are equal, like \( \frac{y}{4} = \frac{4}{y} \). To solve a proportion using cross-multiplication, you follow a straightforward process:

• Identify the two fractions.
• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Do the same with the numerator of the second fraction and the denominator of the first fraction.

For our example, multiply \( y \) (numerator of the first fraction) by \( y \) (denominator of the second fraction) and \( 4 \) (numerator of the second fraction) by \( 4 \) (denominator of the first fraction). This results in the equation \( y^2 = 16 \).
Cross-multiplication is like linking the fractions by their opposite parts. This way, you eliminate the fractions and end up with a simple equation that's easier to solve.
It's crucial to ensure the original fractions are true proportions before cross-multiplying, as doing so only makes sense in such cases.

Square Roots in Equations

A square root is a number which, when multiplied by itself, gives you the original number. For instance, the square root of 16 is 4 because \( 4 \times 4 = 16 \). In algebra, square roots are often used to solve equations like \( y^2 = 16 \).

• To find the square root, use the square root symbol \( \sqrt{} \).
• When dealing with square roots in equations, remember that you generally have two possible solutions: a positive and a negative value.

For \( y^2 = 16 \), when we take the square root of both sides, we get \( y = 4 \) and \( y = -4 \). This is because both \( 4^2 = 16 \) and \( (-4)^2 = 16 \).
Handling square roots becomes second nature with practice. It's important to always consider both the positive and negative solutions unless the context of the problem dictates otherwise.
Remember: not all numbers have real square roots. Positive numbers do, but if you run into negative numbers within a square root, you may need to work with imaginary numbers.

Taking a Closer Look at Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They often involve variables like \( y \) that you solve for using various techniques such as cross-multiplication and taking square roots.

• Algebraic equations can be simple, like \( 2x + 3 = 7 \), or more complex, involving polynomials, exponents, and other mathematical operations.
• Solving an algebraic equation involves finding the value of the variable that makes the equation true.

In the exercise, the equation \( y^2 = 16 \) comes from the cross-multiplication step, and we solve for \( y \) using square roots. In broader terms, solving algebraic equations can require:

• Rearranging terms to isolate the variable.
• Applying operations inversely to balance the equation.
• Simplifying expressions step by step.

Algebraic equations are foundational in mathematics, forming the basis for more advanced topics. Understanding how to manipulate and solve these equations is crucial for progressing to more advanced mathematics topics.