Problem 57 - Algebra for College Students

Question

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. \(0.7 n+1.4=3.92\)

Step-by-Step Solution

Answer

The solution is \(n = 3.6\).

Step 1: Identify the Given Equation

We are given the equation \(0.7n + 1.4 = 3.92\). The task is to solve for \(n\).

Step 2: Isolate the Variable Term

Subtract 1.4 from both sides of the equation to get \(0.7n = 3.92 - 1.4\). This simplifies to \(0.7n = 2.52\).

Step 3: Solve for the Variable

Divide both sides of the equation by 0.7 to find \(n\). Thus, \(n = \frac{2.52}{0.7}\). Using a calculator, \(n\) is found to be 3.6.

Step 4: Verify the Solution

Substitute \(n = 3.6\) back into the original equation to ensure it satisfies the equation: \(0.7 \times 3.6 + 1.4\) should equal 3.92. Calculate \(0.7 \times 3.6 = 2.52\), and \(2.52 + 1.4 = 3.92\). Since both sides equal, our solution is verified.

Key Concepts

Decimal Representation

Decimals play a crucial role in solving linear equations, like the one from our exercise. Here, "decimal representation" simply means expressing numbers with a decimal point. When converting fractions or solving equations with decimals, it is important for the solution to result in a decimal form, especially because decimals are more intuitive for many measurements and contexts.

Here's why decimals are useful:

• Readability: Decimals make numbers more accessible and easier to read.
• Precision: They offer greater precision for calculations, which can be critical in engineering and scientific applications.
• Common Usage: Many real-world applications, including currency and measurements, often require decimal numbers.

In the given problem, solving for \( n \) naturally results in a decimal format, namely \( n = 3.6 \). Formatting results this way ensures clarity, especially when calculators are used to aid in calculations.

Calculator Use in Algebra

In algebra, calculators can be valuable tools, especially when dealing with decimals. They assist in accurate computations, reducing human error significantly. In our exercise, the calculator helped convert the division \( \frac{2.52}{0.7} \) into a neat decimal \( n = 3.6 \).

Using calculators effectively involves:

• Entering equations precisely to avoid mistakes.
• Understanding functions such as addition, subtraction, multiplication, and division with decimals.
• Using parentheses to maintain the correct order of operations.

While calculators can simplify the arithmetic, ensure to double-check each step mentally or on paper to avoid over-reliance. Remember, calculators offer speed, but understanding the "why" behind calculations is key to mastery.

Verifying Solutions

Checking your solutions in algebra ensures that the problem was solved correctly. Verification involves plugging the found solution back into the original equation to see if both sides are equal. This method confirms the accuracy of your answer.

For our exercise, after finding \( n = 3.6 \), verify like so:

• Substitute \( n = 3.6 \) back into the equation: \( 0.7 \times 3.6 + 1.4 \).
• Calculate: \( 0.7 \times 3.6 = 2.52 \).
• Add: \( 2.52 + 1.4 = 3.92 \).
• Compare with the original equation's right side which is 3.92.

Both sides match, so the solution holds true. Verification helps catch any potential errors early, giving confidence in your results.