Problem 40 - Algebra A Combined Function

Question

Solve each equation for \(y .\) $$ x=-6 y $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{x}{-6} \).

Step 1: Identify the Equation Structure

The given equation is a linear equation in one variable: \( x = -6y \). Our objective is to solve for \( y \), finding \( y \) in terms of \( x \).

Step 2: Isolate the Variable

To solve for \( y \), isolate it on one side of the equation. Start by dividing both sides of the equation by \(-6\) to get \( y \) by itself. \[\frac{x}{-6} = \frac{-6y}{-6}\]

Step 3: Simplify the Equation

On the right-hand side, the \(-6\)s cancel out, leaving us with \( y \). This simplifies the equation to: \[y = \frac{x}{-6}\]

Step 4: Write the Final Answer

The equation \( y = \frac{x}{-6} \) is the solution for \( y \) in terms of \( x \).

Key Concepts

Solving Equations

Solving equations is the process of finding the value of unknown variables that satisfy the given mathematical statement. In our example, we work with a linear equation, which is characterized by having variables raised to the power of one. The goal is to express one variable in terms of another. This often requires performing operations like addition, subtraction, multiplication, or division to both sides of the equation. Each operation should maintain the equation's balance. Linear equations can usually be simplified to a form where one variable stands alone on one side of the equation.

Isolation of Variables

Isolation of variables is a crucial step in solving equations. It involves rearranging the equation so that the variable you wish to solve for is on one side. In the given example, the equation is structured to solve for \(y\). We aim to have \(y\) by itself on one side by undoing any other operations affecting it. Typically, this means performing the inverse operation. For instance, if a term multiplies \(y\), you can divide by that term to isolate \(y\). This approach also adheres to the principle of performing the same action on both sides to keep the equation balanced.

Division in Equations

Division in equations plays a vital role in isolating variables or simplifying expressions. When a variable is multiplied by a constant, like in the equation \(x = -6y \), division helps remove that constant. By dividing both sides of the equation by \(-6\), you ensure that \(y\) becomes isolated on one side, simplifying the expression to \(y = \frac{x}{-6}\). It's important to remember that division is the inverse of multiplication, which makes it a powerful tool in solving equations. Always check that division does not involve zero, as division by zero is undefined.