Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$-3 x+4=-1$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{3}\).

Step 1: Isolate the Variable Term

First, we want to get all terms involving the variable on one side of the equation and constant terms on the other side. We start by subtracting 4 from both sides of the equation to isolate the term with \(x\).\[-3x + 4 - 4 = -1 - 4\]This simplifies to:\[-3x = -5\]

Step 2: Solve for the Variable

Now, we need to solve for \(x\). Since \(-3x\) means \(-3\) times \(x\), we divide both sides of the equation by \(-3\) to get \(x\) by itself.\[\frac{-3x}{-3} = \frac{-5}{-3}\]This simplifies to:\[x = \frac{5}{3}\]

Key Concepts

Solving Equations

Equations are like balanced scales. Both sides must remain equal. When solving equations, the goal is to find the value of a variable, often represented as \(x\), that makes the equation true. We do this by performing operations that maintain the balance.

• Begin by simplifying each side of the equation if needed, such as removing parentheses or combining like terms.
• Ensure all terms with the variable are on one side and constants on the other. This often involves adding, subtracting, multiplying, or dividing.
• Perform the same operation on both sides to maintain equality. Be mindful of inverse operations which undo what's being done to the variable.

Solving step-by-step helps eliminate mistakes. It ensures clarity and correct results while making each process straightforward to follow.

Isolation of Variable

When solving equations, one key step is the isolation of the variable. This means getting the variable by itself on one side of the equation. In our example, we isolate \(x\) by moving other terms to the opposite side while keeping the balance of the equation.

• Subtract or Add: Use addition or subtraction to move constants away from the variable. In the equation \(-3x + 4 = -1\), subtracting 4 from both sides helped isolate the \(x\) term.
• Division or Multiplication: If the variable is multiplied by a number, use division to isolate it. If divided, use multiplication.

Moving terms across the equation is like shifting weight on a balance scale; you need to keep both sides equal to maintain correctness.

Step-by-Step Solution

A step-by-step solution improves understanding by breaking down the problem-solving process into manageable parts. This helps to pinpoint errors quickly and make complex equations seem more approachable. Let's unpack the process used in the example step-by-step:

• Identify Required Operations: Identify what needs isolation, in this case, the \(x\). Look at the operations involved, such as addition, subtraction, and division, to determine how to proceed.


• Execute Each Step Rigorously: Take each identified step carefully. Subtract 4 from both sides, then divide by \(-3\). Each operation is critical to isolating the variable.


• Verify Your Answer: After solving, substitute back into the original equation to verify the solution is correct. This ensures the step-by-step approach led to the right results.

Step-by-step solving not only fosters better comprehension but also instills confidence in problem-solving.