Problem 72 - Beginning and Intermediate Algebra

Question

The sum of six-fifths of a number and 2 is 14 .

Step-by-Step Solution

Answer

The number is 10.

Step 1: Define the variable

Let the unknown number be denoted by the variable x.

Step 2: Set up the equation

The sum of six-fifths of a number and 2 can be written as: \( \frac{6}{5}x + 2 \). Given that this is equal to 14, we write the equation: \( \frac{6}{5}x + 2 = 14 \).

Step 3: Isolate the term with the variable

Subtract 2 from both sides of the equation to isolate the term with the variable: \( \frac{6}{5}x + 2 - 2 = 14 - 2 \), which simplifies to \( \frac{6}{5}x = 12 \).

Step 4: Solve for x

Multiply both sides of the equation by the reciprocal of \( \frac{6}{5} \) to solve for x: \( x = 12 \times \frac{5}{6} \).

Step 5: Simplify the solution

Simplify the multiplication to find the value of x: \( x = 12 \times \frac{5}{6} = 10 \).

Key Concepts

solving equations

When solving equations, the goal is to determine the value of the unknown variable that makes the equation true.
Generally, we follow a sequence of steps to achieve this:

• Define the variable.
• Set up the equation based on the problem statement.
• Isolate the term with the variable.
• Solve for the variable.
• Simplify the solution if needed.

Before we dive into these steps, let's emphasize one important thing: balance. Think of an equation like a balance scale. Whatever we do to one side, we must also do to the other side, so the scale remains balanced. This principle is key when isolating the variable and solving for its value.

fractions in equations

Fractions in equations can initially seem daunting, but they're just as manageable with the right approach.
Here’s how to handle them:

• Recognize the fraction: Identify where the fraction appears in the equation. Fractions can represent coefficients of variables or constants.
• Clear the fraction by multiplying: To handle a fraction like \( \frac{6}{5} \), think about the reciprocal (i.e., \( \frac{5}{6} \)). Multiply both sides of the equation by this reciprocal to clear the fraction.

For example, in our equation \( \frac{6}{5}x = 12 \), we multiplied both sides by \( \frac{5}{6} \) to solve for x.

This gives us: \( x = 12 \times \frac{5}{6} \). Remember, multiplying by the reciprocal is a powerful method to simplify and solve equations with fractions, ensuring we maintain balance.

variable isolation

Variable isolation refers to the process of getting the variable alone on one side of the equation.
This step is crucial because it allows us to see what the variable equals. Here’s a breakdown:

• Identify the variable term: In our example, it's \( \frac{6}{5}x \).
• Remove constants: Subtract or add constants from both sides to isolate the variable term. Eg. Subtract 2 from both sides: \( \frac{6}{5}x + 2 - 2 = 14 - 2 \).
• Simplify: Perform the arithmetic to maintain balance.

Once the variable term is isolated, as in \( \frac{6}{5}x = 12 \), we can then move on to solving for the variable by using operations like division or multiplication.

Ensuring we execute these steps carefully and maintaining balance will lead to the correct value of our variable.