Problem 61 - Introductory Algebra for College Students

Question

Use the given information to write an equation. Let x represent the number described in each exercise. Then solve the equation and find the number. The difference between \(\frac{2}{5}\) of a number and 8 is \(\frac{7}{5}\) of that number. Find the number.

Step-by-Step Solution

Answer

The number x in this exercise is -\frac{1}{8}.

Step 1: Formulating the algebraic equation

The problem can be formulated into an equation by following the word description: The difference between \(\frac{2}{5}\) of a number (which can be written as \(\frac{2}{5}x\)) and 8 is equal to \(\frac{7}{5}\) of that same number (which can be written as \(\frac{7}{5}x\)). Therefore, the equation as per the given description becomes: \(\frac{2}{5}x - 8 = \(\frac{7}{5}x\)

Step 2: Solving the algebraic equation

Now, this equation should be solved for \(x\). First, move the term \(\frac{7}{5}x\) to the left side of the equation to collect all terms with \(x\). This gives: \(\frac{2}{5}x - \frac{7}{5}x = 8\). Simplifying it further, -\(\frac{1}{x}\) = 8.

Step 3: Final calculation

Finally, to solve for \(x\), divide both sides by -1. This gives: \(x = -\frac{1}{8}\).

Key Concepts

Understanding Algebraic Equations

Algebraic equations are fundamental in introductory algebra. They are mathematical statements that use an equal sign to show that two expressions are equivalent. The goal is to find the value of the variable that makes the equation true. In the given exercise, we are asked to find a number, represented by the variable \( x \), that fulfills the given condition.

The equation is derived from the problem statement: "The difference between \(\frac{2}{5}\) of a number and 8 is \(\frac{7}{5}\) of that number." This can be translated into an algebraic equation:

• \(\frac{2}{5}x\): Represents the fraction of the number.
• The phrase "difference between \(\frac{2}{5}x\) and 8" becomes: \(\frac{2}{5}x - 8\).
• "Is equal to" indicates where we place the equal sign in the equation: \(=\).
• "\(\frac{7}{5}\) of that number" is expressed as \(\frac{7}{5}x\).

The equation then logically follows as:
\[\frac{2}{5}x - 8 = \frac{7}{5}x\]

Working with Fractions in Algebra

Fractions are common in algebra and they can sometimes make equations seem more complicated. However, understanding how to manipulate them is key to solving these equations. When dealing with fractions in algebra, especially when they involve variables, it's important to know:

• How to combine like terms.
• How to move terms across the equation.

In our example, the equation involves fractions \(\frac{2}{5}x\) and \(\frac{7}{5}x\). To simplify and solve for \( x \), you must collect all terms involving \( x \) on one side of the equation. In this case:
- Move \(\frac{7}{5}x\) to the left side to get: \(\frac{2}{5}x - \frac{7}{5}x\).
- This combines into \(-\frac{5}{5}x\) or simply \(-x\), which we equate to 8.
Pay close attention to the negative sign when handling such expressions. It often signifies that you will need to divide by a negative coefficient, as shown in the next section.

Solving Algebraic Equations

Solving equations in algebra involves finding the value of the unknown variable that makes the equation true. Let's see how this works with the equation we have:

• We have \(-x = 8\) after combining like terms from \(\frac{2}{5}x - \frac{7}{5}x\).
• To isolate \( x \), divide both sides by \(-1\), which gives us \(x = -8\).

It's important to translate each step of your operations back to the equation:
- Moving \(\frac{7}{5}x\) changed the sign of x terms.
- Dividing by \(-1\) flips the sign, giving us our solution.
Checking your work is always a good idea, especially when negative solutions are involved. In our final answer, substituting \( x = -8 \) into the original equation should satisfy its conditions.