Problem 7 - Precalculus

Question

Solve each equation. $$2 x-3=-5$$

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Isolate the variable

To solve for \(x\), we need to isolate it on one side of the equation. Start by adding 3 to both sides of the equation to get rid of the -3: \[ 2x - 3 + 3 = -5 + 3 \] This simplifies to: \[ 2x = -2 \]

Step 2: Solve for x

Now, divide both sides by 2 to solve for \(x\): \[ \frac{2x}{2} = \frac{-2}{2} \] Simplifying gives: \[ x = -1 \]

Key Concepts

Isolating Variables in Linear Equations

When you are faced with a linear equation, such as \(2x - 3 = -5\), the first step towards finding a solution is isolating the variable—here, the variable is \(x\). To do this, you'll need to move all terms involving \(x\) to one side of the equation while keeping all constant terms (numbers without variables) on the other side. In this scenario, start by addressing the \(-3\) term. To get rid of it, you should perform the inverse operation. Since \(-3\) is subtracted from \(2x\), adding \(3\) to both sides helps in cancelling it out.

• Equation: \(2x - 3 = -5\)
• Add 3 to both sides: \(2x - 3 + 3 = -5 + 3\)
• Simplify: \(2x = -2\)

Now, the variable \(x\) is isolated alongside its coefficient. With the variable on one side, you're closer to solving the equation.

Solving Equations by Simplifying

After isolating the variable, the next task is solving the equation by simplifying it. In this example, we have simplified the expression to \(2x = -2\). The equation indicates a multiplication of \(x\) by 2. Therefore, to solve for \(x\), we need to perform the inverse operation. This is known as dividing by the coefficient of \(x\), which in this case is 2.

• Equation: \(2x = -2\)
• Divide each side by 2: \(\frac{2x}{2} = \frac{-2}{2}\)
• Simplify: \(x = -1\)

By simplifying correctly, we've determined that \(x = -1\). Solving equations like this requires careful attention to mathematical operations to ensure each step properly moves towards finding the solution.

Understanding Step-by-Step Solutions

Approaching linear equations with step-by-step solutions helps in understanding how each mathematical operation affects the equation. By breaking down complex processes into individual actions, it becomes easier to follow the steps required to reach the solution.A step-by-step approach ensures:

• Clarity: Each step is evidently explained, which improves comprehension and replicability.
• Accuracy: Focusing on each individual action minimizes mistakes.
• Skill-building: Gradually practicing steps increases problem-solving efficiency.

For example, by reviewing each step needed to solve \(2x - 3 = -5\), learners can internalize:

• How to isolate variables effectively.
• The importance of reversing operations to maintain equation balance.
• The process of simplifying to arrive at the correct answer.

Using these step-by-step solutions as a learning tool boosts confidence and encourages independent problem-solving abilities.