Problem 58 - 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. \(0.14 n-0.26=0.958\)

Step-by-Step Solution

Answer

The solution is \(n = 8.7\).

Step 1: Isolate the variable term

Start by adding 0.26 to both sides of the equation to get rid of the constant term on the left side: \[0.14n - 0.26 + 0.26 = 0.958 + 0.26\]This simplifies to:\[0.14n = 1.218\]

Step 2: Solve for the variable

Divide both sides of the equation by 0.14 to solve for \(n\):\[\frac{0.14n}{0.14} = \frac{1.218}{0.14}\]This simplifies to:\[n = 8.7\]

Step 3: Check the solution

To validate the solution, substitute \(n = 8.7\) back into the original equation and see if both sides are equal:\[0.14(8.7) - 0.26 = 0.958\]Calculate the left side:\[1.218 - 0.26 = 0.958\]Since both sides are equal, \(n = 8.7\) is correct.

Key Concepts

Decimal Form

Understanding decimal form is key when dealing with linear equations involving decimals. Decimal form is a way of writing numbers that utilize a base-ten system, which we often use in everyday life.
Instead of writing fractions, we express numbers with one or more spaces past the decimal point, like 0.14 or 1.218. In calculations, decimal form helps ensure precision and is easier to work with than fractions.
**Why Use Decimal Form?**
Decimals are an integral part of solving equations in real-world scenarios, such as measurements and currency. Being familiar with manipulating decimals can greatly simplify the process of solving equations.
**Tips for Working with Decimals:**

• Always line up the decimal points vertically when adding or subtracting decimals.
• For multiplication, count the total number of decimal places in both numbers to place the decimal in the correct position in the product.
• When converting fractions to decimals, divide the numerator by the denominator.

This understanding facilitates precision, especially helpful in ensuring your final answer is correct and in the required form.

Isolate the Variable

Isolating the variable is a critical step in solving linear equations. It means rearranging the equation such that the variable is by itself on one side of the equation. This allows you to easily identify the value of the variable.
**Steps to Isolate the Variable:**

• Identify the variable you need to solve for.
• Perform inverse operations to remove any constants on the same side as the variable. For example, in the equation \(0.14n - 0.26 = 0.958\), you would add 0.26 to both sides to eliminate the constant \(-0.26\).
• If the variable is multiplied by a coefficient, divide both sides by that coefficient to solve for the variable. For example, dividing both sides by 0.14 will isolate \(n\).

It's essential to perform the same operation on both sides of the equation to keep it balanced. Much like maintaining balance on a seesaw, what you do to one side, you must do to the other.
Mastering this skill is foundational for algebra, enabling you to tackle more complex equations with confidence.

Validate Solution

Validation of a solution is the final step in equation solving and serves as a confirmation that your answer is correct. It involves substituting your solution back into the original equation to ensure both sides are equal.
This not only checks your work but also boosts confidence in your solution.
**Steps for Validation:**

• Take the solution from your equation (like \(n = 8.7\)).
• Substitute it back into the original equation.
• Calculate both sides to check if they are equivalent.

Using our equation example: substituting \(n = 8.7\) results in \(0.14 \times 8.7 - 0.26 = 0.958\), which confirms our solution is correct as both sides equal.
**Why Validate?** Validation is crucial because it ensures that your calculations were correct and that the solution is applicable to the problem scenario. This reduces errors and increases reliability in problem-solving.