Problem 14 - Precalculus Mathematics for Calculus

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. $$3+\frac{1}{3} x=5$$

Step-by-Step Solution

Answer

The solution is x = 6.

Step 1: Eliminate Fractions

Multiply the entire equation by 3 to eliminate the fraction: \[ 3 imes 3 + 3 imes rac{1}{3} x = 3 imes 5 \] This simplifies to: \[ 9 + x = 15 \]

Step 2: Isolate Variable x

Subtract 9 from both sides of the equation to get x by itself: \[ 9 + x - 9 = 15 - 9 \] This simplifies to: \[ x = 6 \]

Key Concepts

Solving Equations with Fractions

Working with equations that contain fractions can be challenging, but with the right strategy, it becomes straightforward. The key is to eliminate fractions from the equation to simplify it. This makes further calculations easier and prevents fractional complications.
To eliminate fractions, find a common denominator of all fractional terms. Multiply every term in the equation by this common denominator. This eliminates the fractions by transforming the equation into one without fractions.
In the example of the equation \(3+\frac{1}{3} x=5\), we eliminate the fraction by multiplying each term by 3, which is the denominator of the fraction. This transforms our equation into \(9 + x = 15\), a much simpler form without fractions.

Isolating Variables

Isolating the variable is an essential technique in solving equations. It means rearranging the equation so that the variable in question stands alone on one side of the equation. This helps to find the value of the variable directly.
To isolate the variable, follow these steps:

• Identify the term containing the variable.
• Use arithmetic operations (addition, subtraction, multiplication, or division) to move other terms to the opposite side of the equation.

In the equation \(9 + x = 15\), to isolate \(x\), we subtract 9 from both sides. This leaves us with \(x = 6\). Now, \(x\) is by itself, providing a clear solution.

Steps in Solving Equations

Solving equations involves a series of methodical steps to simplify and find the solution. Each step has its purpose and brings you closer to solving the equation.
Here's a structured approach:

• Simplify the equation: Start by eliminating any fractions or parentheses by distributing and combining like terms.
• Isolate the variable: Rearrange the equation to get the variable on one side by itself, using inverse operations like addition/subtraction and multiplication/division.
• Check your solution: Substitute the value found back into the original equation to verify correctness.

Using the previous example \(3+\frac{1}{3} x=5\), we first eliminate fractions to get \(9 + x = 15\), then isolate the variable \(x\) to find \(x = 6\). A final check confirms the accuracy of our solution.