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Chapter 2: Problem 206

Solve the following equations with variables and constants on both sides. $$5 y-30=-5 y+30$$

Short Answer

Expert verified
y = 6

Step by step solution

01

- Combine like terms

Start by adding or subtracting the same terms on both sides to simplify the equation. Add \(5y\) to both sides to eliminate \(-5y\) on the right side: \[5y - 30 + 5y = -5y + 30 + 5y\]\ This simplifies to: \[10y - 30 = 30\]
02

- Isolate the variable term

Add 30 to both sides to get the constant on one side: \[10y - 30 + 30 = 30 + 30\]\ This simplifies to: \[10y = 60\]
03

- Solve for the variable

Divide both sides by 10 to solve for \(y\): \[ \frac{10y}{10} = \frac{60}{10} \] This simplifies to: \[ y = 6 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
An algebraic equation is a mathematical statement that shows the equality between two expressions. The expressions include variables (often represented by letters like x or y) and constants (specific numbers). In the exercise provided, we worked with the equation: $$5y - 30 = -5y + 30$$. Understanding how to balance, manipulate, and ultimately solve these equations is critical in algebra. These equations are tools for describing relationships and making predictions in various fields such as science, engineering, and finance.
Isolating the Variable
Isolating the variable means moving all instances of the variable to one side of the equation, leaving the constants on the other side. This helps simplify the process of solving the equation. Let's consider the equation from the exercise:When we add 5y to both sides, we aim to remove the -5y on the right: \(5y - 30 + 5y = -5y + 30 + 5y\). This simplifies to: \(10y - 30 = 30\). By isolating the variable term, we can more easily solve for the variable itself.
Combining Like Terms
Combining like terms is an essential step in simplifying algebraic equations. Like terms refer to terms that have the same variable raised to the same power. In the exercise, the like terms are 5y and -5y. By adding 5y to both sides, we combine these like terms, resulting in: \(5y + 5y = 10y\). Combining like terms simplifies the equation and prepares it for the next steps towards finding the solution.
Solving for Variables
The final goal is to solve for the variable, which means finding the value of the variable that makes the equation true. In the exercise, after isolating the variable and combining like terms, we end at: \(10y = 60\). By dividing both sides by 10, we solve for y: \( \frac{10y}{10} = \frac{60}{10} \), which reduces to \( y = 6 \). This step usually involves basic arithmetic operations such as addition, subtraction, multiplication, or division, to isolate the variable and determine its numerical value.

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