Problem 54 - Elementary Algebra

Question

Solve each of the following equations and express the solutions in decimal form. Your calculator might be of some help. $$-3.2 x-1.6=5.8$$

Step-by-Step Solution

Answer

The solution is \(x \approx -2.3125\).

Step 1: Identify the equation

The given equation is \(-3.2x - 1.6 = 5.8\). Our goal is to solve for \(x\).

Step 2: Isolate variable terms

Add 1.6 to both sides of the equation to start isolating terms involving \(x\):\[-3.2x - 1.6 + 1.6 = 5.8 + 1.6\]This simplifies to:\[-3.2x = 7.4\]

Step 3: Solve for \(x\)

Divide both sides by \(-3.2\) to solve for \(x\):\[x = \frac{7.4}{-3.2}\] Calculate this using a calculator:\[x \approx -2.3125\]

Step 4: Verify the solution

Substitute \(x = -2.3125\) back into the original equation to confirm it satisfies \(-3.2x - 1.6 = 5.8\):Calculate:\(-3.2(-2.3125) - 1.6 = 5.8\). This verifies that our solution is correct.

Key Concepts

Decimal Solutions

When you've got a linear equation and need to find a solution, converting the equation into a decimal form is often the final step. Decimal solutions provide precise answers that are easy to understand and use in further calculations.
Linear equations can, of course, be solved to yield exact fractions or whole numbers. However, sometimes a decimal gives a more practical view of the answer, especially when using calculators.
Here are a few pointers about working with decimal solutions:

• Decide early if a decimal approximation is appropriate for your situation. Calculators are excellent for this task, providing quick and accurate results.
• When expressing a fraction as a decimal, remember that some decimals repeat. Round off as necessary based on the precision required, but be cautious of rounding errors.
• In practical applications like measurement or finance, decimal precision can be crucial. Always know the level of precision that meets your needs.

With these fundamentals in mind, you've got a clear path to precise, workable solutions in linear equation problems.

Isolation of Variables

The essence of solving linear equations is to isolate the variable—here, represented as \(x\). This means re-arranging the equation until \(x\) is all by itself on one side of the equation.
The process requires undoing any addition or subtraction first, followed by division or multiplication. Here’s a structured way to understand it:

• Look at the equation and identify terms associated with the variable. Also, identify constants—numbers without variables.
• Add or subtract constants from both sides to remove them. This turns the equation into a simpler form.
• If multiplication or division by a number impacts the variable, do the opposite operation, allowing the variable to stand alone.

Ensuring each step is carefully executed guards against errors, ensuring the solution is accurate. Practice these steps so they become second nature, allowing you to handle increasingly complex equations confidently.

Step-by-Step Problem Solving

Breaking down math problems into manageable steps is the heart of solving equations efficiently. Each step moves you closer to the solution, simplifying and tackling different parts individually.
For the problem at hand, a step-by-step approach consists of the following:

• Recognize the entire equation. Knowing what you're dealing with right from the start sets a clear goal.
• Follow the isolation of variables. Gradually performing operations on both sides simplifies the work.
• Calculate the solution step and verify. Once you have an answer, substitution back into the original problem confirms the accuracy.

Such problem-solving skills are not only limited to math; they form a critical part of approaches in various disciplines. Practice makes perfect, so repeat and practice until you feel comfortable with linear equations. Start developing a strategic mindset that helps handle equations methodically and creatively.