Problem 20 - Algebra Form and Function

Question

Solve the equations. $$ \frac{x}{x-4}=\frac{2}{3} $$

Step-by-Step Solution

Answer

Rational Equation: $$\frac{x}{x-4}=\frac{2}{3}$$ Answer: There is no solution to this rational equation.

Step 1: (Step 1: Find a common denominator)

(In this case, the common denominator is the product of the two denominators: (x-4) and 3.)

Step 2: (Step 2: Multiply both sides of the equation by the common denominator)

(Multiply both sides of the equation by the common denominator, (x-4)*3, to clear the fractions.) $$ (x-4)3\cdot\frac{x}{x-4}=(x-4)3\cdot\frac{2}{3} $$

Step 3: (Step 3: Simplify the equation)

(Simplify the equation by canceling out the denominators.) $$ x(3)=2(x-4) $$

Step 4: (Step 4: Distribute and solve for x)

(Now that we have no more fractions in the equation, we can distribute and solve for x.) $$ 3x=2x-8 $$ $$ x=-8 $$

Step 5: (Step 5: Check the solution against the original equation)

(Plug the value of x back into the original equation to ensure it's a valid solution.) $$ \frac{-8}{-8-4}=\frac{2}{3} $$ Since the left side of the equation becomes -4/3 after substituting x with -8, and this is not equal to the right side of the equation, there is no solution for this rational equation.

Key Concepts

Common Denominators

In rational equations, a common denominator helps eliminate fractions and simplifies solving. Let's simplify this concept by thinking about solving simple fractions in math.
Imagine you have two fractions with different denominators. To add or compare them, you need a common denominator to proceed.

• Identify each fraction's denominator.
• Find a number that each denominator can divide evenly. This is your common denominator.
• Multiply each part of your equation by this common denominator to clear the fractions.

In the equation \( \frac{x}{x-4} = \frac{2}{3} \), the common denominator is \((x-4) \times 3\). Multiplying both sides by this expression turns the equation into something with no fractions, easing the solving process.

Solving Equations

Once we've eliminated fractions by using a common denominator, the next step is solving the equation. This often means simplifying and performing arithmetic operations to isolate the variable you're solving for.

• Start by distributing any factors outside parentheses in your expression.
• Combine like terms on both sides of the equation if applicable.
• Use addition, subtraction, multiplication, or division to isolate the variable.

In our specific example, the equation is simplified from \(x(3) = 2(x-4)\) to \(3x = 2x - 8\). We can further simplify by subtracting \(2x\) from both sides to get \(x = -8\). However, sometimes isolating the variable might not always yield a valid solution.

Checking Solutions

This is an essential step often overlooked. Checking ensures our solution satisfies the original equation and highlights any errors.

• Plug your solution back into the original equation; replace the variable with your answer.
• Perform the calculations on both sides.
• Verify if both sides of the equation are equal.

In the example \( \frac{x}{x-4} = \frac{2}{3} \), substituting \(x = -8\) yields \( \frac{-8}{-8-4} = \frac{-8}{-12} = \frac{2}{3} \). After calculating, we find out \(-\frac{2}{3} eq \frac{2}{3}\), indicating our solution is not valid. Thus, it concludes there is no solution for the initial equation.