Problem 47 - Prealgebra

Question

Solve. $$6+3 y=4$$

Step-by-Step Solution

Answer

The solution is \(y = -\frac{2}{3}\).

Step 1: Isolate the Variable Term

Start by moving the constant from the left side of the equation to the right side. Subtract 6 from both sides of the equation: \[6 + 3y - 6 = 4 - 6\]This simplifies to: \[3y = -2\].

Step 2: Solve for the Variable

Now, solve for \(y\) by dividing both sides by 3: \[\frac{3y}{3} = \frac{-2}{3}\]This simplifies to: \[y = -\frac{2}{3}\].

Key Concepts

Isolation of Variables

In solving linear equations, isolating the variable is often a crucial first step. Imagine an equation as a balance scale, where both sides must remain equal. The concept of isolating a variable involves rearranging the equation so that the variable of interest is by itself on one side of the equation. This often requires moving other terms to the other side.
In our original exercise, the equation is \(6 + 3y = 4\). The goal is to isolate \(y\). To start, we need to remove the constant from the left side. We achieve this by performing the same operation on both sides of the equation. Subtract 6 from both sides:

• \(6 + 3y - 6 = 4 - 6\)

Now the equation is simplified to \(3y = -2\).
This operation maintains the equality of the equation, ensuring that the scale remains balanced while isolating the variable term.

Simplification of Equations

Simplification is the process of making an equation easier to work with while keeping its core meaning intact. By breaking down and combining like terms, we can transform a complex equation into a form that's straightforward to interpret and solve.
In our exercise, once the variable term \(3y\) was isolated, the equation became \(3y = -2\). Here, the equation was simplified by removing the constant from the left side. Simplification ensures that we can clearly see the relationship between the variables and constants involved.
The goal is to get the equation in the simplest form which is often needed for intuition and for further solving operations. As a general tip, aim to clear unnecessary operations or components that may complicate the steps ahead.

Solving for a Variable

Solving for a variable means finding the value of a variable that makes the equation true. After isolating and simplifying, the equation \(3y = -2\) is ready for us to solve for \(y\).
Here's how we do it: Divide both sides of the equation by the coefficient of \(y\), which is 3. This allows us to extract the value of \(y\) alone:

• \(\frac{3y}{3} = \frac{-2}{3}\)

This results in \(y = -\frac{2}{3}\).
By performing the same operation on both sides, we ensure the equation remains balanced. Finding the value of a variable is often the final step in solving an equation, and it confirms our understanding of the relationship expressed in the equation.