Problem 6 - Elementary Algebra

Question

Solve. $$33 x-1+4=5$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{33}.\)

Step 1: Simplify the Equation

First, simplify the left side of the equation by combining like terms. Start with \(33x - 1 + 4\). Combine \(-1\) and \(+4\) to get \(3\). This gives us: \[33x + 3 = 5.\]

Step 2: Isolate the Variable Term

Subtract \(3\) from both sides of the equation to move the constant term to the right. This gives us: \[33x + 3 - 3 = 5 - 3,\] which simplifies to: \[33x = 2.\]

Step 3: Solve for the Variable

Next, divide both sides of the equation by \(33\) to solve for \(x\). This gives us: \[x = \frac{2}{33}.\]

Key Concepts

Simplifying Expressions

Simplifying an expression is about making it as basic as possible. In our exercise, we start with an equation like this: \(33x - 1 + 4 = 5\). We need to combine the constants on the left side, namely \(-1\) and \(+4\). When we add these two numbers, we get \(3\). Rewriting the expression with this simplified form looks like this: \(33x + 3 = 5\). This is a crucial first step because it reduces the clutter, making the equation easier to handle. Just remember:

• Look for numbers you can combine.
• Add or subtract constants.
• Get to the simplest form.

Simplifying expressions sets the stage for the next steps of solving equations.

Isolating Variables

Isolating the variable is like solving a mystery where you need to find out exactly what a certain letter or symbol represents. In this case, we're solving for \(x\). Once we have the simplified expression \(33x + 3 = 5\), the next logical step is to get \(x\) by itself on one side of the equation. This involves moving the constant \(3\) from the left side by subtracting it from both sides of the equation: 1. Start with \(33x + 3\) on the left and \(5\) on the right.2. Subtract \(3\) from both sides to cancel out the \(+3\): \(33x + 3 - 3 = 5 - 3\).This leaves us with: \(33x = 2\). Now, the variable \(x\) stands alone on the left, making it easy to solve for it in the next step. Always aim to:

• Move constants to the opposite side of the equation.
• Use addition or subtraction wisely to keep it balanced.

Combining Like Terms

Combining like terms involves gathering similar types of numbers or variables together within an expression. This is an extension of simplifying an expression, and it is crucial for clarity. In our step-by-step solution, we noticed terms like \(-1\) and \(+4\) used in the same expression \(33x - 1 + 4\). These are both simple number terms.Though our given exercise doesn't have multiple terms for the variable \(x\), combining like terms often involves doing the same for variables too. Let's say we had \(5x + 3x\), we would combine these to get \(8x\).Some tips for combining like terms:

• Identify coefficients of the same variable.
• Add or subtract coefficients as needed.
• Simplify constants separately, as shown in the problem.

Understanding how to combine like terms helps keep equations tidy and manageable, as we focus on solving them effectively.