Problem 29 - College Algebra

Question

For the following exercises, solve for the variable. $$ 4 y+8=2 y $$

Step-by-Step Solution

Answer

The solution is \(y = -4\).

Step 1: Identify the Equation

The given equation is \(4y + 8 = 2y\). Our goal is to solve for \(y\).

Step 2: Isolate the Variable Terms

Move all the terms involving \(y\) to one side of the equation. Subtract \(2y\) from both sides to get \(4y - 2y + 8 = 0\).

Step 3: Simplify

Simplify the expression by combining like terms, resulting in \(2y + 8 = 0\).

Step 4: Isolate the Variable

Subtract 8 from both sides of the equation to isolate the variable term: \(2y = -8\).

Step 5: Solve for the Variable

Divide both sides by 2 to solve for \(y\): \(y = \frac{-8}{2}\).

Step 6: Simplify the Solution

Simplify \(y = \frac{-8}{2}\) to get \(y = -4\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential technique used to solve equations and simplify expressions. In algebra, we often have to rearrange equations to find the value of a variable, just like in this exercise. The primary goal is to perform operations on the equation without altering its equality.
This process involves:

• Addition or subtraction of the same number on both sides to maintain equality.
• Multiplying or dividing both sides of the equation by the same non-zero number.

In our exercise, we subtracted 2y from both sides in an initial step to gather all 'y' terms together, simplifying the manipulation of the expression. This action respects the equation's balance, aiding further simplification.

Isolating Variables

Isolating variables is a crucial skill for solving equations. This involves rearranging the equation so that one side contains only the variable, while all other terms are on the opposite side. The purpose is to find the exact value of the variable.
In the exercise, to isolate the variable term involving 'y', we performed the following steps:

• After combining like terms, move constants away from the variable term by subtracting 8 from both sides. This gives us the simpler form, 2y = -8.
• Finally, we divide each side by 2 to completely isolate 'y'.

This clear border between the variable and numbers facilitates finding the variable's value. It's all about keeping equations neat and manageable.

Combining Like Terms

Combining like terms is an essential step in simplifying equations, allowing us to streamline complex expressions into simpler ones. Like terms are terms that have the same variable raised to the same power. Only the coefficients of these terms can differ.
In our example equation, 4y and 2y are like terms, both containing the variable 'y'. By subtracting 2y from 4y, we simplified the equation to 2y + 8 = 0.

• This procedure reduces clutter, making the equation easier to solve.
• It's important to be careful with signs when combining like terms to avoid mistakes.

Emphasizing the similarities among terms aids in quick computation and comprehension.

Simplifying Expressions

Simplifying expressions is an effective way to break down and solve algebraic problems. The goal is to reduce an expression or equation to its simplest or most manageable form.
During our exercise, simplification involves multiple actions:

• Combining like terms to condense the expression.
• Performing basic arithmetic operations such as subtraction and division.

For instance, after combining terms, we subtracted 8 to isolate the variable term and then divided by 2 to express 'y' in its simplest form, y = -4. This step helps clarify the path to the variable's solution, ensuring we reach accurate, succinct results.