Problem 64 - Elementary Algebra Within Reach

Question

Isolating the Variable Is it okay to isolate the variable on the right side of the equation? Illustrate your answer using the equation \(11=3 x+2\).

Step-by-Step Solution

Answer

Yes, it is okay to isolate the variable on the right side of the equation. Following the steps of subtraction and division, the solution to the equation \(11 = 3x + 2\) is \(x = 3\).

Step 1: Identify the Equation

The equation provided in the problem is \(11 = 3x + 2\). The goal is to solve for \(x\).

Step 2: Isolate the Variable

Start by isolating \(x\) on one side of the equation. To do this, subtract \(2\) from both sides of the equation to maintain balance. \(11 - 2 = 3x + 2 - 2\) simplifies to \(9 = 3x\).

Step 3: Solve for the Variable

To fully isolate \(x\) on the right side of the equation, divide both sides by \(3\). \(9 / 3 = 3x / 3\) simplifies to \(x = 3\) which is the solution to the equation.

Key Concepts

Isolating the Variable

Isolating the variable is a crucial step in solving equations as it helps us find the specific value that satisfies the equation. It means getting the variable, in our case \(x\), alone on one side of the equation. This process allows us to determine the exact value needed to balance both sides of the equation.

When isolating a variable, it's perfectly acceptable to have it on either the left or right side of the equation. The main goal is to simplify the equation until the variable stands alone. For example, in the equation \(11 = 3x + 2\), we begin isolating \(x\) by removing constants or coefficients that are with \(x\). This usually involves operations like addition, subtraction, multiplication, or division.

With practice, you'll find that isolating the variable becomes a straightforward and almost automatic process in solving linear equations.

Equation Balance

Maintaining balance in an equation is like keeping a scale even. Whatever operation you perform on one side of the equation, you must also perform on the other side. This ensures the equality holds true.

Let's see this concept in action with our equation \(11 = 3x + 2\). To start the process of isolating \(x\), we subtract \(2\) from both sides to maintain balance: \(11 - 2 = 3x + 2 - 2\). This simplifies to \(9 = 3x\).

Next, to fully isolate \(x\), we divide both sides by \(3\): \(9 / 3 = 3x / 3\). This division again maintains balance, ultimately resulting in \(x = 3\). Every step must be balanced so the equality is never disrupted. This approach ensures the solution remains valid and accurate.

Algebraic Manipulation

Algebraic manipulation involves using fundamental operations such as addition, subtraction, multiplication, and division to rearrange equations. This toolkit allows us to simplify and solve equations methodically.

Let's break down the manipulation steps for our equation \(11 = 3x + 2\):

• First, we subtract \(2\) from both sides, simplifying the equation to \(9 = 3x\).
• Next, we use division to handle the coefficient of \(x\). Dividing both sides by \(3\) gives us \(x = 3\).

With each manipulation step, the equation becomes simpler. The goal is to isolate \(x\), making it clear what \(x\) equals.

Algebraic manipulation is like navigating through steps to achieve clarity and resolve the unknown factor in an equation. Patience and practice are key, and soon these manipulations become intuitive, paving the way to mastering equations.