Problem 48 - Prealgebra

Question

Solve. $$8+3 y=4$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = -\frac{4}{3}\).

Step 1: Isolate the variable term

The given equation is \(8 + 3y = 4\). To begin solving for \(y\), we need to isolate the term with \(y\). Subtract 8 from both sides to cancel the 8 on the left side. This gives us: \[3y = 4 - 8\].

Step 2: Simplify the equation

Perform the subtraction on the right side of the equation: \[4 - 8 = -4\]. The equation now simplifies to: \[3y = -4\].

Step 3: Solve for \(y\)

To isolate \(y\), divide both sides of the equation by the coefficient of \(y\), which is 3: \[y = \frac{-4}{3}\]. This gives us the value of \(y\).

Key Concepts

Isolating Variables

The first step in solving a linear equation like \(8 + 3y = 4\) is to focus on isolating the variable, which in this case is \(y\). Think of isolating a variable as making sure that you have the variable on one side of the equation by itself. To do this, you need to get rid of any other terms that are sitting on the same side as your variable.
In our exercise, this means removing the \(8\) from the left side, where the \(3y\) is. We do this because it cleans up the equation and allows us to work directly with \(y\).
To remove the \(8\), you subtract \(8\) from both sides of the equation. This subtraction "cancels" the \(8\), moving it to the other side and leaving you with \(3y = 4 - 8\).
Using subtraction on both sides is crucial because it keeps the equation balanced. It's like a see-saw that needs to stay level; whatever you do on one side, you must do on the other.

Solving Equations

Now that you have isolated \(3y\) on the left side of the equation, comes the next step which falls under the core concept of solving equations. The new expression is \(3y = 4 - 8\).
First, highlight simplifying the right side by performing the subtraction. Calculate \(4 - 8\) which is \(-4\). Now, your equation looks more straightforward: \(3y = -4\).
Solving equations is much like unraveling a mystery—you are figuring out what number the variable represents. In this step, we're closer, but not quite there yet. We need to deduce the value of \(y\) without the \(3\) next to it. This brings us to our next step in solving this equation completely.

Simplifying Expressions

Simplifying expressions involves making them clearer and easier to understand or work with. In the context of our equation, simplifying is needed to find the exact value of \(y\).
Considering \(3y = -4\), simplifying here means dividing both sides by \(3\) to get \(y\) alone. The division by \(3\) is effectively reducing the expression. After the division, you end up with the simplest form of the solution: \(y = \frac{-4}{3}\).

• This shows the power of simplification.
• It allows you to make sense of equations.
• Helps find the value of unknown variables more easily.

Simplifying is the final polishing step in solving any equation, allowing for a clear and neat result.