Problem 17 - Algebra and Trigonometry

Question

Contain linear equations with constants in denominators. Solve each equation. $$\frac{x}{3}=\frac{x}{2}-2$$

Step-by-Step Solution

Answer

The solution to the equation \( \frac{x}{3} = \frac{x}{2} - 2 \) is \(x = 12\).

Step 1: Clearing the fractions

Find the least common multiple of the denominators, 2 and 3, which is 6. Multiply each term of the equation by 6. \n \(6 * \frac{x}{3} = 6 * \frac{x}{2} - 6*2\), which simplifies to \(2x = 3x -12\).

Step 2: Rearranging the equation

Rearrange the equation in order to isolate 'x'. We can do this by subtracting \(3x\) from both sides of the equation. This gives us \(-x = -12\).

Step 3: Solving for x

The equation can be easily solved now by multiplying both sides by -1 to get the value of 'x'. Thus, \(x = 12\).

Key Concepts

Clearing Fractions in Equations

When you encounter a linear equation with fractions, clearing those fractions can simplify the equation and prepare it for easier solving. To clear fractions, you find a common number that each denominator can multiply into evenly - this is called the Least Common Multiple (LCM). For instance, if you have \(\frac{x}{3} = \frac{x}{2}-2\), the denominators are 3 and 2.

By determining the LCM of 3 and 2, which here is 6, you can multiply each term by 6. This eliminates the fractions, because \(6\times\frac{1}{3}\) and \(6\times\frac{1}{2}\) are whole numbers. Clearing fractions is a critical first step in solving linear equations because it allows you to work with integers, simplifying subsequent steps.

Finding Least Common Multiple

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers. It is a key concept in clearing fractions from equations. To find the LCM, list the multiples of each number until a common multiple appears. For example, the multiples of 2 are 2, 4, 6, 8, etc., and the multiples of 3 are 3, 6, 9, etc. The first common multiple we see is 6, making it the LCM of 2 and 3.

Finding the LCM helps in solving equations because once you've multiplied each term by the LCM, the denominators will cancel out, leaving an equation with no fractions, which is generally easier to solve.

Isolating Variables

Once fractions are cleared, isolating the variable is the next step. This means rearranging the equation so that the variable you are solving for is on one side of the equation by itself. To isolate the variable \(x\) in the simplified equation \(2x = 3x - 12\), subtract \(3x\) from both sides to get a new equation, \(2x - 3x = -12\), which simplifies further to \( -x = -12\).

By isolating the variable, you turn the problem into a basic one-step algebraic equation that can be easily solved, bringing you closer to finding the value of the variable.

Linear Equation Step-by-Step Solution

The final reconciliation of variables in a linear equation involves a step-by-step process to provide the solution. After isolating the variable and having \( -x = -12\), you’ll now solve for \(x\). Since \(x\) is multiplied by -1, divide both sides by -1, effectively changing the signs to get \( x = 12\).

Every step is crucial to ensure that you simplify the equation methodically and avoid errors. Then, by confirming your result, you've successfully employed a systematic approach, leveraging your knowledge of fractions, multiples, and algebraic manipulations to resolve the equation with clarity and precision.