Problem 37 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ x-\frac{1}{3} x-\frac{1}{2} x-5=0 $$

Step-by-Step Solution

Answer

The solution is \(x = 30\).

Step 1: Combine Like Terms

The given equation is \(x - \frac{1}{3}x - \frac{1}{2}x - 5 = 0\). First, combine the like terms that involve \(x\). This involves combining the coefficients:\[1 - \frac{1}{3} - \frac{1}{2}\]

Step 2: Find a Common Denominator

To combine the fractions, find a common denominator for \(1\), \(\frac{1}{3}\), and \(\frac{1}{2}\). The common denominator is 6. Thus, express each term with denominator 6: \(1 = \frac{6}{6}\), \(\frac{1}{3} = \frac{2}{6}\), \(\frac{1}{2} = \frac{3}{6}\).

Step 3: Subtract the Fractions

Now subtract the fractions to combine them:\[\frac{6}{6} - \frac{2}{6} - \frac{3}{6} = \frac{1}{6}\]Thus, the combination of coefficients becomes \(\frac{1}{6}x\).

Step 4: Rewrite the Equation

Insert the combined coefficient back into the equation:\[\frac{1}{6}x - 5 = 0\]

Step 5: Solve for \(x\)

Add 5 to both sides of the equation to isolate the term with \(x\):\[\frac{1}{6}x = 5\]Then multiply both sides by 6 to solve for \(x\):\[x = 30\]

Key Concepts

Combining Like Terms

When solving linear equations, one of the first steps is often to simplify the equation by combining like terms.Combining like terms means to add or subtract terms that have the same variable raised to the same power.This makes the equation simpler and easier to solve.

In the given equation:

• We have three terms involving the variable \( x \): \( x \), \( \frac{1}{3}x \), and \( \frac{1}{2}x \).
• Since all these terms involve the same variable \( x \), they are considered like terms.
• We need to combine them by adding or subtracting their coefficients.

To do this, first list their coefficients: 1 for \( x \), \(-\frac{1}{3} \) for \( \frac{1}{3}x \), and \(-\frac{1}{2} \) for \( \frac{1}{2}x \).
Combining these coefficients involves handling fractions correctly, which is explained in the next sections on fractions and common denominators.

Solving Equations

The goal when solving equations is to find the value of the variable that makes the equation true.Linear equations, such as the given one, have terms with a variable raised to the power of one.

To solve the equation after combining like terms, the equation simplifies to form:

• An immediate next step is to isolate the term containing the variable.
• In case of the equation \(\frac{1}{6}x - 5 = 0\), add 5 to both sides to get \(\frac{1}{6}x = 5\).

Here, multiplying both sides by 6 helps to isolate \( x \) completely:

• It simplifies to \( x = 30 \), which is your solution.
• This process often involves inverse operations, like addition, subtraction, multiplication, or division, to isolate the variable.

Being familiar with these operations is key to effectively solving any linear equation.

Fractions and Common Denominators

Fractions can make solving equations tricky, but a common denominator helps simplify things.A common denominator is a shared multiple of all denominators, allowing us to combine fractions more easily.

To combine fractions, find a common denominator as follows:

• Identify the denominators in your fractions.
• In \(1\), \(\frac{1}{3}\), and \(\frac{1}{2}\), use 6, the smallest common multiple of 3 and 2.

Rewrite each term using the common denominator:

• This transforms the coefficients to \( \frac{6}{6} \), \( \frac{2}{6} \), \( \frac{3}{6} \).
• Finally, perform the subtraction: \( \frac{6}{6} - \frac{2}{6} - \frac{3}{6} = \frac{1}{6} \).

Handling fractions by finding a common denominator is a crucial skill, simplifying the operation and clearing the path to solve for the variable.