Problem 34 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solutions. $$ \frac{x-3}{x}=\frac{x}{x+6} $$

Step-by-Step Solution

Answer

The equation has no solution. Check shows that no x-value can be substituted into the original equation to get a valid equality.

Step 1: Clear the Fractions

To clear the fractions, multiply every term by the least common denominator (LCD). The LCD of all the denominators, 'x' and 'x+6', is \(x(x + 6)\). So the equation becomes: \[(x-3)(x + 6) = x^2\]

Step 2: Simplify and Rearrange the Equation

Expand the left hand side and subtract \(x^2\) from both sides to bring all terms on one side of the equation, which gives: \(x^2 - 3x + 3x + 18 - x^2 = 0\). Simplifying this, it becomes: \(18 = 0\)

Step 3: Solve Equation and Check Solution

Note that in this case, after simplification, we ended up with an inconsistent equation, \(18 = 0\). This means this equation has no solutions.

Key Concepts

Solving Equations

Solving algebraic equations is about finding the value of the unknown variable that makes the equation true. Generally, equations involve two sides – the left-hand side and the right-hand side, equated by an "=" sign. When solving, the goal is to simplify both sides and isolate the variable on one side. This often involves operations like addition, subtraction, multiplication, and division to handle the terms and simplify the equation.

When dealing with fractions in an equation, it's common to find a common denominator or use elimination techniques to simplify the equation. Make sure to check if the final equation keeps balanced by verifying the solution.

• Begin with simplifying expressions and moving terms.
• Utilize operations to isolate the variable.
• Double-check the solution by substituting back into the original equation.

Checking is crucial because it ensures that the derived solution truly works within the original equation constraints.

Clear Fractions

Equations featuring fractions can appear intimidating, but clearing fractions simplifies them significantly. The technique involves eliminating fractions to work with whole numbers, making the equation easier to manage.

To clear fractions, multiply every term in the equation by the least common denominator (LCD). This transforms fractional equations into polynomial ones. For instance, consider the equation \( \frac{x-3}{x} = \frac{x}{x+6} \). Finding the LCD, which is \( x(x+6) \), you multiply every term by it. The equation then transitions from containing fractions to:

• \((x-3)(x+6) = x^2\)

This step drastically simplifies equation handling, making it a straightforward algebraic expression. Always remember that every term needs adjustment during multiplication, ensuring nothing is left partially handled.

Least Common Denominator

The Least Common Denominator (LCD) is the smallest multiple that is common between the denominators of fractions in an equation. Identifying and using the LCD is essential to clearing fractions effectively, as it allows consolidation of terms, leading to a simpler problem.

When you encounter fractions with different denominators within one equation, identify the LCD to make calculations neat and easy. For example, if an equation involves fractions with denominators \(x\) and \(x+6\), the LCD is \(x(x+6)\). Multiplying every term by this LCD clears the fractions by reconstituting them into whole numbers. This step plays a key role in transforming and reducing the equation down to manageable factors. Ensuring that the LCD is correctly identified and applied is critical, as it maintains the original equivalence of the equation post-transformation.