Problem 22 - Fundamentals of Mathematics

Question

Solve each equation. $$ 4 y-8-6 y=3 y+1 $$

Step-by-Step Solution

Answer

The solution is \(y = -\frac{9}{5}\).

Step 1: Simplify Each Side of the Equation

The goal is to combine like terms on each side of the equation. Start with the left-hand side where you have the terms \(4y - 8 - 6y\). Combine the terms involving \(y\):\[4y - 6y = -2y\]So, the left side becomes \(-2y - 8\). The right side \(3y + 1\) remains the same.

Step 2: Move Variable Terms to One Side

We rearrange the equation to have all the terms with \(y\) on one side. Add \(2y\) to both sides to eliminate \(y\) from the left side:\[-2y - 8 + 2y = 3y + 1 + 2y\]Simplifying gives:\[-8 = 5y + 1\]

Step 3: Isolate the Constant Term

Subtract 1 from both sides to move the constant term to one side of the equation:\[-8 - 1 = 5y + 1 - 1\]Simplifying gives:\[-9 = 5y\]

Step 4: Solve for the Variable

Divide both sides of the equation by 5 to solve for \(y\):\[\frac{-9}{5} = y\]Thus, \(y = -\frac{9}{5}\).

Key Concepts

Equations

When dealing with algebra, one of the fundamental tasks is to solve equations. Equations are like statements that assert two expressions are equal. They contain variables, which are symbols that stand in for unknown values. In an equation like \(4y - 8 - 6y = 3y + 1\), the goal is to find the value of the variable \(y\) that makes this statement true. Solving equations involves using a variety of techniques to isolate the variable and determine its value.

Solving Equations

Solving equations is all about finding the value of the variable that satisfies the equation. The process often involves several steps.

• Simplification: This means combining like terms or simplifying each side of the equation, as seen in the initial step where terms including \(y\) were combined.
• Rearrangement: Adjust the equation so that all terms with variables are on one side, and constants are on the other.
• Isolation of terms: Carefully move terms by adding or subtracting from both sides to simplify the equation further.

The ultimate aim is to manipulate the equation into a form where the variable is isolated and easily solvable.

Variable Isolation

Variable isolation is a key step where you manipulate the equation to get the variable by itself on one side of the equation. This is crucial as it allows you to directly find the value of the unknown. In the given example, isolating \(y\) required manipulating the terms

• First, adding \(2y\) to both sides to get rid of \(y\) from the left side.
• Subtracting 1 from both sides to handle the constant terms.

Ultimately, these steps enabled the equation to be simplified to \(-9 = 5y\), making the problem straightforward to solve for \(y\).

Combining Like Terms

Before solving the equation, it's essential to simplify each side by combining like terms. Like terms are terms that involve the same variable raised to the same power. Consider the original left-hand side of the equation \(4y - 8 - 6y\). Here:

• \(4y\) and \(-6y\) are like terms because they both contain the variable \(y\).
• Combining these gives \(-2y\), simplifying the expression considerably.

This simplification reduces the equation's complexity, making it easier to solve. It's a foundational step in solving many algebra problems effectively.