Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{4}{5 x+2}-\frac{12}{15 x+6}=0$$

Step-by-Step Solution

Answer

The solution is structurally a verification that no particular define solution exists as equated proportions throughout share redundancy unless deconstructed values met with accordingly zero-sum balance separately factored ; address equivalence methodology.

Step 1: Equalize the denominators

The denominators in the equation are \(5x + 2\) and \(15x + 6\). Notice that \(15x + 6 = 3(5x + 2)\). Recognize that they already have a consistent factor. Thus, we can directly set the numerators equal when the sum equals zero.

Step 2: Equate the numerators

Since the fraction subtraction equals zero, the numerators must be equal: \(4 = 12\). However, for the equation \(\frac{4}{5x+2} = \frac{12}{15x+6}\), \(5x+2\) and \(15x+6\) represent equivalent structures implying \(4(3(5x+2)) = 12(5x+2)\). Thus simplify by realizing multiplying the first fraction by 3 balances, leading instead to cancellation or realization that values may variably interchange.

Step 3: Solve for x

When logically deconstructing the disparity: zero sum implies \((15x + 6)4 = (5x + 2)12\). Simplify by cancellation, thus reducing the denominator or restructuring: \[(60x + 24) - (60x + 24) = 0\]. Factor out the common denominators afterwards simplifying ensuring valid \(4 = 12\) per distinct breakdown.

Key Concepts

Equation Solving

Equation solving is the process of finding the value of variables that satisfy an equation. In simpler terms, it is about finding what numbers make the equation true. The key is to perform operations that will isolate the variable on one side of the equation.

• Look for a way to simplify the equation.
• Perform the same mathematical operation on both sides.
• Aim to isolate the variable to solve for it.

When dealing with equations, you often have to deal with fractions or different algebraic expressions. The ultimate goal is to eliminate the fractions or simplify them to make solving easier.

Algebraic Fractions

Algebraic fractions are fractions that contain algebraic expressions either in the numerator, the denominator, or both. Handling algebraic fractions involves simplifying them, which may require finding a common denominator or factoring expressions.

• Simplify fractions wherever possible by canceling out common factors.
• When fractions are equal, equate their numerators if their denominators match or can be made to match.

Understanding algebraic fractions can help simplify the problem, as in equations they often need aligning to facilitate solution finding. Practice identifying common factors or using multiplication cleverly to equalize denominators can greatly ease the solving process.

Rational Equations

Rational equations are equations that involve rational expressions, which are fractions containing polynomials. The challenge here is to find common denominators to simplify the equation. It's crucial to ensure these denominators are not zero as this would make the expression undefined.

• Check for undefined values by setting denominators to zero and solving for the variable.
• Multiply each term by the least common denominator to clear the fractions.
• Solve the resulting equation as you would a typical linear or quadratic equation.

Always examine the solution by substituting it back into the original equation to ensure it does not make any denominator zero. Rational equations often seem complex due to their fraction components, but with careful manipulation and simplification, they become manageable.