Question

\(\frac{y}{x}=8\)

Short answer

The value of y in terms of x is y = 8x

Step-by-step solution

Isolation of y

To isolate y, multiply both sides of the equation by x. This cancels out the x in the denominator on the left side. Your equation turns into: y = 8x

Key concepts

Equations

In mathematics, equations are fundamental tools that help express relationships between different quantities. An equation is a statement that asserts the equality of two expressions, typically involving variables and constants. For example, in the given equation \(\frac{y}{x}=8\), both sides are balanced, meaning that whatever value the expressions take, they will always be equal when the condition is satisfied.
Equations come in various forms such as linear, quadratic, or polynomial, each serving a different purpose in mathematics and real-life applications. Solving an equation often means finding the values of the variables that make the equation true. It's an essential skill in algebra and is foundational for advancing in math.

Variables

Variables are symbols, often represented by letters like \( x \) and \( y \), used to denote unknown or changeable values in equations or other mathematical expressions. They serve as placeholders that can take on various numerical values. In our example \(\frac{y}{x}=8\), \( x \) and \( y \) are variables.
Understanding how variables work is crucial because they allow mathematicians and students to create generalizations and find solutions to problems. Variables open up the exploration of 'what-if' scenarios and are integral in forming and solving equations. They bring flexibility and dynamic into mathematics by allowing equations to represent a wide range of situations.

Solving Equations

Solving an equation involves finding the value of the variables that satisfy the equation's conditions. In our specific example, the goal is to solve the equation \(\frac{y}{x} = 8\) by identifying what \( y \) should be, given any value of \( x \).
To solve equations, one often relies on various methods such as substitution, elimination, or using algebraic manipulation to balance the equation. Methods can vary based on the complexity of equations, from simple arithmetic management to applying advanced theorems and techniques in mathematics. Solving equations allows us to find answers and make informed decisions in diverse fields, from engineering to economics.

Isolation Techniques

Isolation techniques are strategies used in algebra to get a single variable alone on one side of an equation. This makes it easier to solve for that variable. The key idea is to maintain equality by performing the same operation on both sides of the equation. In the original exercise \(\frac{y}{x} = 8\), we isolated \( y \) by multiplying both sides by \( x \). This results in the equation \( y = 8x \).

• First, identify the variable you want to isolate.
• Use operations likes addition, subtraction, multiplication, or division to move other terms or numbers away from the variable.
• Perform each operation equally on both sides to maintain the balance of the equation.

This approach is critical for simplifying complex equations and is a fundamental technique in algebra. Mastering isolation techniques makes solving equations much more intuitive and helps in breaking down even the most challenging problems into manageable steps.