Problem 23 - Elementary Algebra

Question

\(x-3 y=9\) for \(x\)

Step-by-Step Solution

Answer

The solution is \(x = 9 + 3y\).

Step 1: Identify the Equation

The given equation is \(x - 3y = 9\). We need to solve for \(x\).

Step 2: Isolate the x-term

To isolate the \(x\) on one side, add \(3y\) to both sides of the equation: \(x - 3y + 3y = 9 + 3y\).

Step 3: Simplify the Equation

This simplifies to \(x = 9 + 3y\), which is the expression for \(x\) in terms of \(y\).

Key Concepts

Isolation of Variables

The technique "isolation of variables" refers to rearranging an equation so that one specific variable stands alone on one side of the equation. This is an essential step in solving linear equations. When we isolate a variable, we're essentially determining what value, or expression in terms of other variables, it equates to. For instance, in the equation \(x - 3y = 9\), to solve for \(x\), we want \(x\) to be by itself on one side of the equation.
To achieve this, we perform algebraic manipulations, such as addition or subtraction, to eliminate other terms from the side of the \(x\). In this case, we add \(3y\) to both sides to isolate \(x\). This process ensures that the variable is separated from other variables and constants, which simplifies further analysis or computations.
By mastering isolation of variables, we can systematically approach complex equations and solve them efficiently.

Simplifying Equations

After isolating the variable, the next step involves simplifying the equation. Simplifying an equation means making it as straightforward as possible. This process often includes

• Combining like terms
• Reducing fractions
• Simplifying expressions involving distribution

In the example \(x = 9 + 3y\), the equation is already simplified once we have isolated \(x\). There are no like terms to combine or fractions to reduce, making it an immediate expression of \(x\) in terms of \(y\).
Simplification is crucial as it enhances clarity and makes an equation easy to interpret and use in further computations or discussions. It also helps in checking work for accuracy, ensuring that the final expression is as direct as possible.

Algebraic Manipulation

Algebraic manipulation is the process of using algebraic operations to transform and solve equations. This includes operations such as addition, subtraction, division, and multiplication, all applied strategically to achieve the desired form or solution.
In the equation \(x - 3y = 9\), algebraic manipulation allows us to isolate \(x\) by recognizing that adding \(3y\) to both sides will cancel out the \(-3y\) on the left, simplifying our task. This manipulation builds on fundamental algebraic properties, such as the equality principle which states that the equality remains the same

if you perform the same operation on both sides of an equation

Mastering algebraic manipulation empowers students to approach more complicated equations with confidence, as it develops strategic thinking in breaking down and reconstructing mathematical statements. By practicing these skills, students enhance their problem-solving toolkit.