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Chapter 2: Problem 28

Multiple Choice Solve \(2 y+3=9\). (Algebra Review)

Short Answer

Expert verified
The solution is \( y = 3 \).

Step by step solution

01

Isolate the term with the variable

Subtract 3 from both sides of the equation to move the constant to the right side. This will help isolate the term with the variable.\[2y + 3 - 3 = 9 - 3\]This simplifies to:\[2y = 6\]
02

Solve for the variable

Now, divide both sides of the equation by 2 to solve for \(y\).\[\frac{2y}{2} = \frac{6}{2}\]This simplifies to:\[y = 3\]

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Key Concepts

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

Understanding Linear Equations
Linear equations are one of the simplest types of equations you'll encounter in algebra. They typically involve variables raised only to the first power. In other words, there are no squares, cubes, or other powers. A standard form of a linear equation looks like \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. These constants can be any real number. The variable is simply represented by \( x \) or any other letter.
  • Linear equations can usually be visualized as straight lines when graphed on the coordinate plane.
  • They have consistent rates of change, meaning the graph's slope remains the same.
Learning how to handle them is essential, as they form the foundation for understanding more complex types of equations.
The Process of Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. To do this, we often follow a set of systematic steps known as algebraic processes. Let's break them down:
  • Isolate the Variable: This first step involves getting the term with the variable by itself on one side of the equation. In our example, \( 2y + 3 = 9 \), subtracting \( 3 \) from both sides was necessary to isolate \( 2y \).
  • Simplification: The next step is to simplify both sides of the equation. With the isolated \( 2y \), we checked our work, leading to \( 2y = 6 \).
  • Solve for the Variable: Finally, divide both sides by \( 2 \) to solve for \( y \). This yields \( y = 3 \), which satisfies the equation.
Each step ensures clarity and accuracy, leading you from the original equation to the solution.
Algebraic Manipulation Techniques
Algebraic manipulation encompasses a variety of techniques to rearrange equations into more usable forms. These techniques let us move constants and variables across the equation while maintaining equality. Some common strategies include:
  • Adding or Subtracting: This involves adding or subtracting the same value from both sides to move terms around. In our exercise, subtracting \( 3 \) was the first manipulation to isolate \( 2y \).
  • Multiplying or Dividing: We often multiply or divide both sides to handle coefficients attached to variables. For example, dividing by \( 2 \) simplified our result to \( y = 3 \).
  • Distribution: This is useful when dealing with expressions inside parentheses, although not needed for our current example.
These methods are fundamental skills in algebra, enabling you to transform equations into simpler, more solvable forms. Mastery of these techniques creates a strong foundation for tackling not only linear equations but a wide range of algebraic problems.

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