Question

Solve the equation by using the LCD. Check your solution(s). $$\frac{x-3}{x-4}+4=\frac{3 x}{x}$$

Short answer

The solution to the given equation is \(x = \frac{7}{2}\).

Step-by-step solution

Identify the Denominators in the Problem

In this equation, the denominators are \(x\), \(x-4\), and \(x\). Any expression that has \(x\) or \(x-4\) as factor would work as a LCD.

Find the LCD (Least Common Denominator)

The least common denominator (LCD) for the denominators \(x\) and \(x-4\) is \(x(x-4)\).

Multiply the Equation by the LCD

Multiplying the equation by the LCD \(x(x-4)\) will eliminate the fractions. So, \(x(x-4)\frac{x-3}{x-4} +x(x-4)4 = x(x-4)\frac{3 x}{x}\) simplifies to \(x(x-3) +4x(x-4)=3x(x-4)\).

Simplify the Equation

Upon simplification, the equation becomes \(x^2 -3x +4x^2 -16x =3x^2-12x\). Collecting like terms and simplifying, we get \(2x^2 -7x =0\).

Solve the Simpler Equation

The above equation can be solved by using the zero-product property. First, factor the equation: \(x(2x-7) = 0\). Set each factor equal to zero and solve for x. So, \(x=0\) or \(2x -7 =0\) which gives \(x = \frac{7}{2}\).

Check the Solutions

Substitute the solutions in the original equation to check if they hold true. It is seen that \(x=0\) does not satisfy the original equation as it makes the denominator zero. So, the only correct solution is \(x = \frac{7}{2}\).

Key concepts

Least Common Denominator

Understanding the concept of the least common denominator (LCD) is essential in solving equations involving fractions. Fractions can be challenging because they have denominators, which are the numbers below the fraction line. To simplify the addition or subtraction of fractions, it's vital to make their denominators the same, and this is where the LCD comes into play.

The LCD is the smallest number that all the denominators in a set of fractions can divide into perfectly. It's like finding a common ground where all these fractions can easily interact. To solve an equation with fractions using the LCD:

• Identify the unique denominators.
• Find the LCD by determining the least common multiple of these denominators.
• Multiply every term of the equation by the LCD to clear the fractions, resulting in a simpler equation.

For example, with denominators of x and (x-4), the LCD will be their product x(x-4) because neither denominator is a multiple of the other. This approach ultimately turns a complex fractional equation into a standard polynomial equation, streamlining the solving process.

Zero-Product Property

The zero-product property is a critical algebraic property that states if the product of two factors equals zero, then at least one of the factors must also be zero. In mathematical terms, if ab = 0, then either a = 0, b = 0, or both. This property is especially useful when solving quadratic equations after factoring.

To apply this property effectively, you should:

• Bring all terms to one side of the equation to set it equal to zero, if it's not already.
• Factor the equation as much as possible.
• Apply the zero-product property by setting each factor equal to zero and solving for the variable.

For the provided exercise, we used the property by first factoring to obtain x(2x - 7) = 0. Then, set each factor equal to zero to find the potential solutions, x = 0 or 2x - 7 = 0. This is a logical and systematic way to uncover the solutions to a polynomial equation.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental process in solving equations. It involves reducing complex expressions into their simplest form, making them easier to understand and solve. Simplification often includes combining like terms, factoring, expanding expressions, and canceling common factors in fractions.

To simplify effectively, follow these guidelines:

• Combine like terms, which are terms that have the same variable raised to the same power.
• Look for opportunities to factor expressions.
• Eliminate any terms that have a factor of zero.
• Reduce any complex fractions by finding a common factor in the numerator and denominator and canceling it out.

In the context of the exercise, we combined the x-terms and x²-terms to end up with the equation 2x² - 7x = 0. Factoring and simplifying gets us to the core of the equation, revealing the most straightforward route to finding the solution.