Problem 66 - Elementary Algebra

Question

Solve. $$ 33-x=33 $$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Understand the Equation

The given equation is \(33 - x = 33\). This equation includes a single variable \(x\) and a constant on both sides of the equation.

Step 2: Isolate the Variable

To find the value of \(x\), we need to isolate it. In this equation, we will start by moving \(33\) from the left side of the equation. We can do this by subtracting 33 from both sides: 33-x-33=33-33.

Step 3: Simplify the Equation

After subtracting, the equation simplifies on both sides: 1. The left side becomes \(33 - 33 - x = -x\). 2. The right side becomes \(33 - 33 = 0\). Thus, the equation simplifies to \(-x = 0\).

Step 4: Solve for x

Now that we have \(-x = 0\), to solve for \(x\), multiply both sides by -1 to get \(x = 0\).

Key Concepts

Solving Linear Equations

Linear equations are equations of the first degree, which means they involve only the first power of the variable. They generally have the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The goal in solving these equations is to find the value of \( x \) that makes the equation true. When solving a linear equation, follow these steps:

• Identify the terms on both sides of the equation.
• Use arithmetic operations to eliminate constants and coefficients until the variable is isolated.
• Ensure the equation remains balanced by performing the same operation on both sides.

This process leads to a simplified equation, making it easier to find the value of the variable.

Isolation of Variables

The isolation of variables is a crucial step in solving algebraic equations. It involves manipulating the equation until the variable is alone on one side. This enables clear identification of its value. In the given problem, \( 33 - x = 33 \), our target is to isolate \( x \). Here's how this is done:

• Subtract 33 from both sides to remove the constant from the left side. This gives you \( 33 - 33 - x = 33 - 33 \).
• The equation simplifies to \(-x = 0\).

Isolation is essential because it directly leads to determining the solution of the variable. It is the step that directly precedes the final step of solving for the variable.

Simplification of Equations

Simplifying equations means reducing them to their simplest form. This typically involves combining like terms, eliminating constants, or resolving any arithmetic expressions involved. For our specific equation, \( 33 - x = 33 \), simplification occurs as follows:

• After subtracting 33 from both sides, the equation reduces to \(-x = 0\).
• Finally, to isolate \( x \), multiply both sides by -1, resulting in \( x = 0 \).

Simplification makes the equation easier to interpret and solve, clearing away unnecessary complexity. It is a fundamental aspect of algebra that ensures equations are efficiently and correctly solved.