Problem 17 - Algebra and Trigonometry

Question

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: Simplify Both Sides

Start by simplifying both sides of the equation if necessary. In this equation, there are no simplifications needed because both sides are already simplified.

Step 2: Move Variable Terms to One Side

Subtract \(x\) from both sides of the equation to bring all \(x\) terms to one side:\[ x - 3 - x = 2x + 6 - x \]This simplifies to:\[ -3 = x + 6 \]

Step 3: Isolate the Variable

Subtract 6 from both sides to isolate the variable \(x\):\[ -3 - 6 = x + 6 - 6 \]This simplifies to:\[ -9 = x \]

Step 4: Verify the Solution

Substitute \(x = -9\) back into the original equation to verify the solution:\[ -9 - 3 = 2(-9) + 6 \]Simplify both sides:\[ -12 = -18 + 6 \]Both sides equal \(-12\), confirming that \(x = -9\) is the correct solution.

Key Concepts

Solving Equations

In the world of mathematics, solving equations is like unlocking doors to find unknown treasures. An equation is like a balance scale. Whatever you do on one side of the scale, you must do on the other side to keep it balanced. This balance is fundamental when solving equations.

The process involves finding the value of the unknown variable that makes the equation true. Consider the equation given: - \[ x - 3 = 2x + 6 \] Your main goal is to find the value of \( x \) that will make both sides of the equation equal.

The principle here is to perform operations that simplify the equation without changing its equality. This can include adding, subtracting, multiplying, or dividing both sides by the same number. The ultimate aim of solving equations is to find the exact value of the variable, which in this case, turned out to be \( x = -9 \). Simplicity is key, and each step you take will bring you closer to uncovering that unknown treasure.

Algebraic Manipulation

Algebraic manipulation refers to the art of rearranging equations to unveil the mystery of the unknown variable. Think of it as skillfully moving puzzle pieces until they form a complete picture.

To effectively manipulate an equation, you should be comfortable with operations such as addition, subtraction, and multiplication. In our exercise, algebraic manipulation was necessary to combine like terms and simplify the equation.
Imagine you have some puzzle pieces on each side of your equation. For example:- \[ x - 3 = 2x + 6 \] Moving terms between the two sides helps to consolidate all like puzzle pieces on one side. In the exercise:

• Subtract \( x \) from both sides to bring similar terms together.
• Combine like terms.

This manipulation gradually simplifies the equation, eventually leading towards 'solving' it by revealing the latent variable.

Variable Isolation

Variable isolation is like clearing the stage for the star performer in a play. This concept focuses on keeping the variable all by itself on one side of the equation, so you can clearly see its value.

In the exercise, isolating the variable \( x \) was achieved through several operations:1. - Subtract \( x \) from both sides to get rid of the variable on the left side. 2. - The new form is: \[ -3 = x + 6 \]. Now, aim to isolate \( x \). 3. - Subtract 6 from both sides to fully isolate \( x \).The resultant equation becomes: - \[ -9 = x \]This step provides a clear revelation of the solution. It allows you to see the star of the equation, \( x = -9 \), shining brightly and standing alone.