Problem 13 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ x-3=2 x+6 $$

Step-by-Step Solution

Answer

The solution is \( x = -9 \).

Step 1: Move Variables on One Side

To solve the equation, you'll want to get all the terms with the variable on one side of the equation. Start by subtracting \( x \) from both sides:\[x - 3 - x = 2x + 6 - x\]This simplifies to:\[-3 = x + 6\]

Step 2: Isolate the Variable

Now, isolate \( x \) by moving the constant term to the other side. Subtract 6 from both sides:\[-3 - 6 = x + 6 - 6\]This simplifies to:\[-9 = x\]

Step 3: Write the Final Solution

The equation is solved for \( x \), and the solution is:\[x = -9\]

Key Concepts

Solving Equations

Solving an equation involves finding the value of the variable that makes the equation true. In our example, the equation is linear, meaning it forms a straight line when plotted on a graph and involves only one variable with a constant change. A linear equation like this typically takes the form of \( ax + b = c \). Here, our goal is to figure out the value of \( x \).
To do this, follow these simple steps:

• Identify the linear equation: In our problem, it's \( x - 3 = 2x + 6 \).
• Move all terms involving the variable \( x \) to one side. This step is crucial for making the equation easier to solve.
• Perform arithmetic operations to isolate the variable (which we'll talk about more in detail soon).
  

Solving equations is like solving a puzzle. It requires simplifying the equation step by step until the variable is alone on one side of the equation and you can easily see its value.

Isolation of Variables

Isolation of variables means re-arranging the equation to get the variable of interest by itself. This is a key step in solving equations because it allows us to determine exactly what value the variable (in this case, \( x \)) must take to satisfy the equation. In our task, we started by moving terms with the variable to one side. To isolate \( x \), we followed these steps:

• Subtracted \( x \) from both sides, resulting in \(-3 = x + 6\).
• Next, we subtracted 6 from both sides to completely isolate \( x \) on one side of the equation.

After this process, we arrived at a simple expression \(-9 = x\).
Isolation is all about simplifying your work. By focusing on moving other numbers or variables away from the one you're solving for, you make it stand out like a piece of a puzzle waiting to be placed.

Algebraic Manipulation

Algebraic manipulation is the process of using arithmetic operations to rearrange and simplify an algebraic equation. This involves addition, subtraction, multiplication, and division to move numbers and variables across the equation. It requires a good understanding of the properties of equality, which state that what you do to one side of an equation, you must also do to the other to maintain balance.
In our example, the algebraic manipulation involved:

• Subtracting \( x \) from both sides, eliminating variable from one side.
• Subtracting 6 from both sides to move the constant away from \( x \).

These steps ensured \( x \) was isolated, which is the ultimate goal when you manipulate terms in equations. Remember, the key to mastering algebraic manipulation is practice and familiarity with how numbers and variables can be shifted around while maintaining the equation's integrity.