Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ \frac{x}{5}-\frac{x}{2}=3 $$

Step-by-Step Solution

Answer

The solution to the given equation is \( x = -10 \).

Step 1: Clear the Fractions

Firstly, to simplify the equation, remove the fractions. The LCM of the denominators \(2\) and \(5\) is \(10\). Therefore, multiplying every term by \(10\) gives: \(10 * (x/5) - 10 * (x/2) = 10 * 3\) which simplifies to \(2x - 5x = 30\).

Step 2: Solve the Equation

Consolidate like terms on one side of the equals sign to get \( -3x = 30\). Divide each side of the equation by -3, the coefficient of \(x\), to isolate \(x\). Hence, \(x = 30 / -3\).

Step 3: Verify the Solution

Check whether the obtained value of \( x \) when substituted back into the original equation satisfies the equation. For \( x = -10 \), the left side becomes \(-10/5 - (-10)/2 = -2+5=3\), which is indeed the right side.

Step 4: Write the Final Solution

After verifying that the obtained solution works when substituted back into the original equation, it can be confirmed that \(x = -10\) is the solution to the equation.

Key Concepts

Solving Linear Equations

Linear equations are equations where the variable is raised only to the first power. They usually have the form \(ax + b = c\). In the provided exercise, the equation is slightly more complex due to the presence of fractions, but it still represents a linear equation because the variable \(x\) appears only in the first degree.

To solve a linear equation, the goal is to isolate the variable on one side of the equation. This typically involves combining like terms and using inverse operations such as addition, subtraction, multiplication, or division. By following these steps, the complexity of handling different terms or coefficients can be simplified.

In our exercise, the steps to clear fractions and simplify the equation before isolating the variable are crucial to finding the solution efficiently.

Fractions in Equations

Fractions can make solving equations seem tricky, but they can be easily managed by using a strategic approach. Whenever you see fractions in an equation, a good step is to eliminate the fractions by finding a common denominator.

In the exercise, the denominators are \(5\) and \(2\), so the least common multiple (LCM) is \(10\). By multiplying every term in the equation by \(10\), we transform the equation into integer coefficients. This results in \(2x - 5x = 30\).

This elimination simplifies the work needed and brings the equation to a form that is easier to solve. It is essential to handle fractions diligently to ensure no errors in the transformation of the equation.

Verifying Solutions

After finding a potential solution for an equation, it is essential to verify that the solution satisfies the original problem. This step is crucial to ensure accuracy and correctness.

To verify the solution, substitute the solved value back into the original equation. If both sides of the equation balance or are equal, the solution is confirmed. For instance, in the exercise, substituting \(x = -10\) back into the original equation \(\frac{x}{5} - \frac{x}{2} = 3\), results in \( -2 + 5 = 3 \).

Verification is an important last step in solving any mathematical equation. It assures that no mistakes were made in calculations, ensuring the integrity of the solution.