Problem 39 - Algebra A Combined Function

Question

Solve each equation for \(y .\) $$ x=3 y $$

Step-by-Step Solution

Answer

\(y = \frac{x}{3}\)

Step 1: Isolate the Variable

In the equation \(x = 3y\), our goal is to solve for \(y\). To do this, we need to isolate \(y\) on one side of the equation. We can achieve this by dividing both sides of the equation by 3.

Step 2: Simplify the Equation

After dividing both sides by 3, the equation becomes \(y = \frac{x}{3}\). This step ensures all operations are performed correctly to isolate \(y\).

Key Concepts

Isolating Variables

Isolating variables is a fundamental technique in solving linear equations. It involves manipulating an equation so that the variable we are interested in is alone on one side of the equation. For example, in the equation \( x = 3y \), if we want to solve for \( y \), it must be isolated. This usually requires us to perform inverse operations.

To isolate \( y \), we look at the equation and identify what is attached to \( y \). In this case, \( y \) is multiplied by 3. To cancel out this multiplication, we do the opposite operation: divide.

• Divide both sides of the equation by the number 3.
• This leaves \( y \) by itself, as \( x/3 = y \).

It is crucial to apply the same operation to both sides of the equation to maintain equality. This concept helps form the foundation for finding solutions to equations more efficiently.

Simplifying Equations

Simplifying equations is the process of rewriting an equation using simpler or more compact expressions without changing its meaning or solution. This process often makes it easier to solve the equation or further analyze it.

In our example, after isolating the variable \( y \) by dividing both sides by 3, we simplified the equation to \( y = \frac{x}{3} \). The simplification process can include:

• Combining like terms.
• Reducing fractions.
• Clearing out brackets or grouping symbols.

The key is to ensure accuracy in operations; every action simplifying the equation must adhere to mathematical laws to maintain balance and correctness.

Algebraic Manipulation

Algebraic manipulation includes the various techniques we use to rearrange equations and expressions. The goal is often to make equations simpler or to solve them for a specific variable. It can involve a mixture of operations such as addition, subtraction, multiplication, division, and the application of distributive or associative laws.

In the equation \( x = 3y \), algebraic manipulation is crucial because it involves dividing both terms by 3 to isolate \( y \). Here's what the process may look like:

• Identify the operation connecting the variable in question (multiplication by 3 in this case).
• Use division, the inverse operation, to eliminate the number 3 from the side with the variable.

Algebraic manipulation combines logical and methodical approaches to modifying equations, ensuring steps lead to a valid solution without altering the inherent equality of the expressions.