Problem 64 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(y-9=2\)

Step-by-Step Solution

Answer

The solution to the equation is \(y=11\).

Step 1: Identify the given equation

The given equation to solve is \(y-9=2\)

Step 2: Isolate \(y\) on one side

To isolate \(y\) on one side, add \(9\) to both sides of the equation. Doing this, we get the new equation \(y=2+9\)

Step 3: Solve for \(y\)

Adding \(2\) and \(9\) together gives us \(y=11\).

Key Concepts

Isolate Variables

The process of isolating variables is critical in solving algebraic equations. To isolate a variable means to separate the variable from other terms in the equation so that it stands alone on one side of the equation sign. This is done through a series of operations that are applied equally to both sides to maintain the equality balanced.

For instance, in an equation like \(y - 9 = 2\), the variable \(y\) is not isolated because it is grouped with \(-9\). To isolate \(y\), you would perform the inverse operation of what's being done to \(y\), which is adding \(9\), to both sides of the equation. After doing this, you end up with \(y = 2 + 9\), which simplifies to \(y = 11\).

This process is essential in solving linear equations and the foundational concept in algebra that allows us to find the values of unknown quantities.

Basic Algebra

Basic algebra encompasses the fundamental principles and operations used to solve equations. These include operations like addition, subtraction, multiplication, and division, along with knowledge of how to manipulate algebraic expressions and equations.

In the context of solving the equation \(y - 9 = 2\), we use the principle of adding the same number, in this case, \(9\), to both sides of the equation to maintain equilibrium or balance. This move falls within the 'addition property of equality', which is a vital concept in algebra.

Understanding these operations and principles helps one navigate through more complex algebraic problems. It is important to practice these basics to ensure a solid foundation in algebra, setting the stage for more advanced mathematics.

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different variables and constants. A linear equation, which is a type of algebraic equation, features variables that are raised only to the first power and do not multiply each other.

In the exercise \(y - 9 = 2\), the equation is linear because the variable \(y\) is to the first power, and there is a constant \(-9\). Upon solving this equation, as outlined in the step-by-step solution, the goal is to find the value of \(y\) that makes the statement true.

Learning to solve such equations is crucial as they are used to model real-world situations and solve practical problems. The ability to manipulate and solve algebraic equations efficiently is a competency that serves well in various fields that require quantitative reasoning and problem-solving skills.