Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ \frac{15}{x}-4=\frac{6}{x}+3 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 9\).

Step 1: Simplify the equation

Since the goal is to isolate \(x\), a common strategy is to clear the fractions from the equation. First, we need to bring all terms involving \(x\) to one side. Start by grouping the terms with \(x\): \[\frac{15}{x}-\frac{6}{x}=4-3 \]

Step 2: Simplify the left and right side of the equation

On the left side, combine the fractions: \[\frac{(15-6)}{x}=\frac{9}{x} \] On the right side, subtract 4 and 3 to get 1:\[\frac{9}{x} = 1 \]

Step 3: Solve for \(x\)

Now the goal is to isolate \(x\). To do this, simply multiply both sides by \(x\) and then divide by 1:\(x = \frac{9}{1} = 9\)

Key Concepts

Clearing Fractions from Equations

Dealing with fractions in algebra can sometimes be a challenge, but there is a straightforward technique to simplify such matters: clearing fractions from equations. This process involves eliminating all fractions so that you're left with a simpler, equivalent equation.

To clear fractions, find the least common denominator (LCD) of all the fractions present in the equation and multiply each term of the equation by this number. In the context of the exercise \frac{15}{x}-4=\frac{6}{x}+3\, the LCD is \(x\), since that’s the only denominator in the equation. By multiplying every term by \(x\), we effectively remove fractions:

• \((\frac{15}{x})\times x = 15\)
• \(- 4\times x = - 4x\)
• \((\frac{6}{x})\times x = 6\)
• \(3\times x = 3x\)

After multiplying, we're ready for the next phase: combining like terms and further solving the equation. This step simplifies the equation significantly, making it easier to isolate the variable and find the solution.

Isolating Variables

Isolating the variable is a foundational skill in algebra, crucial for solving equations. It involves manipulating the equation so the variable you're solving for is on one side of the equation all by itself.

In our exercise, after clearing the fractions, we need to isolate \(x\). To achieve this, we perform operations that undo the math around the variable. We look for inverse operations, like doing subtract where there is add, or divide where there is multiply. Here's how it’s done step by step for our given equation
\[\frac{9}{x} = 1\]

• Multiply both sides by \(x\) to get \(9 = x (1)\)
• Simplify to discover \(x = 9\)

This results in the variable standing alone, giving us the solution \(x = 9\). The key is to perform the same operation on both sides of the equation to keep it balanced. Remember that whatever you do to one side of the equation, you must do to the other to maintain equality.

Simplifying Expressions

Simplifying expressions is the process of reducing complexity while maintaining the original value. This can involve combining like terms, factoring, expanding expressions, or reducing fractions.

In the step-by-step solution provided, simplifying begins by combining like terms on each side of the equation separately. On the left, we combine \(\frac{15}{x}\) and \(-\frac{6}{x}\) to get \(\frac{9}{x}\). On the right, we recognize that the terms 4 and 3 are constants and their difference is 1, giving us the much simpler equation \(\frac{9}{x} = 1\).

Simplifying expressions makes it easier to see the path to the solution. It can transform a daunting equation into a more approachable problem, which is particularly helpful with more complex algebraic fractions or expressions. Always look for opportunities to simplify before attempting to solve, as this will often save time and reduce potential errors in the process.