Problem 27 - Algebra and Trigonometry

Question

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 equation is given as \( X - \frac{1}{3} X - \frac{1}{2} X - 5 = 0 \). Let's start by combining the like terms involving \(X\). These terms are \(X\), \(-\frac{1}{3}X\), and \(-\frac{1}{2}X\).

Step 2: Find a Common Denominator

The coefficients of \(-\frac{1}{3}X\) and \(-\frac{1}{2}X\) have different denominators (3 and 2). The least common denominator for 3 and 2 is 6.

Step 3: Rewrite Coefficients with Common Denominator

Rewrite \(-\frac{1}{3}X\) as \(-\frac{2}{6}X\) and \(-\frac{1}{2}X\) as \(-\frac{3}{6}X\).

Step 4: Combine the Terms

Now, combine the terms: \(X - \frac{2}{6}X - \frac{3}{6}X = (1 - \frac{2}{6} - \frac{3}{6})X\). This simplifies to \((\frac{6}{6} - \frac{2}{6} - \frac{3}{6})X = \frac{1}{6}X\).

Step 5: Simplify the Equation

Substitute the combined term back into the equation: \(\frac{1}{6}X - 5 = 0\).

Step 6: Solve for X

To solve \(\frac{1}{6}X - 5 = 0\), add 5 to both sides: \(\frac{1}{6}X = 5\). Then, multiply both sides by 6 to solve for \(X\): \(X = 30\).

Key Concepts

Combining Like Terms

When working with linear equations, an important step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In our exercise, we have the terms \( X \), \(-\frac{1}{3}X \), and \(-\frac{1}{2}X \) which are all like terms since they all contain the variable \( X \).
To combine these, we need to first consider their coefficients. In the linear equation \( X - \frac{1}{3}X - \frac{1}{2}X - 5 = 0 \), the coefficients of these terms need to be added together. By focusing only on the coefficients, we can merge these terms effectively. This process simplifies our equation, making it easier to solve.

Least Common Denominator

In mathematics, when you have fractions with different denominators, finding a common denominator helps to simplify the addition or subtraction of those fractions. This technique is crucial in equations with fractional coefficients.
In our example, the coefficients \(-\frac{1}{3} \) and \(-\frac{1}{2} \) need a least common denominator (LCD) to be combined. The smallest number that both 3 and 2 divide into is 6, which is our least common denominator.
Once the LCD is determined, we can rewrite the fractions so that they have this common denominator. For instance, \(-\frac{1}{3}X \) becomes \(-\frac{2}{6}X \) and \(-\frac{1}{2}X \) becomes \(-\frac{3}{6}X \). Now, these terms can be easily added or subtracted since they share a common denominator.

Equation Simplification

The final steps in solving linear equations often involve simplification. After combining like terms and rewriting with a least common denominator, you're left with a simpler equation. From our exercise, we simplified to \(\frac{1}{6}X - 5 = 0\).
To continue, remove constant terms from one side of the equation. In this case, add 5 to both sides, giving \(\frac{1}{6}X = 5\). Afterward, eliminate fractional coefficients by multiplying through by 6, the denominator. This isolates \(X\) and results in \(X = 30\).
Simplification allows for more straightforward solutions, leading to the final answer. Ensuring the equation is in its simplest form helps in accurately determining the variable's value. This step-by-step simplification is essential for mastering the solution of linear equations.