Problem 6 - Precalculus: Functions and Graphs

Question

Solve the equation. \(\frac{5 x+2}{10 x-3}=\frac{x-8}{2 x+3}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{1}{17}\).

Step 1: Cross Multiply

To eliminate the fractions, use the cross multiplication method. This involves multiplying the numerator of one fraction by the denominator of the other and setting them equal, resulting in the equation: \((5x + 2)(2x + 3) = (x - 8)(10x - 3)\).

Step 2: Expand Both Sides

Now expand each side of the equation. First, expand the left side: \((5x + 2)(2x + 3) = 10x^2 + 15x + 4x + 6 = 10x^2 + 19x + 6\).Next, expand the right side: \((x - 8)(10x - 3) = 10x^2 - 3x - 80x + 24 = 10x^2 - 83x + 24\).

Step 3: Combine Like Terms

Write the expanded equation from Step 2 and move all terms to one side of the equation to set it to zero: \(10x^2 + 19x + 6 = 10x^2 - 83x + 24\).Combine like terms to get: \((19x + 83x) + (6 - 24) = 0\) which simplifies to \(102x - 18 = 0\).

Step 4: Solve for x

Now, solve the simplified equation \(102x - 18 = 0\) by adding 18 to both sides: \(102x = 18\).Next, divide both sides by 102: \(x = \frac{18}{102}\).Simplify the fraction \(x = \frac{1}{17}\).

Key Concepts

Cross Multiplication

Cross multiplication is a technique commonly used to solve equations involving fractions. The idea is to multiply both sides of an equation by the denominators of the fractions, effectively eliminating the divisions.
This method is particularly handy when you have a proportion, which is an equation involving two equal ratios.

Here's how it works with our equation:

• First, identify the numerators and denominators: for our equation \(\frac{5x + 2}{10x - 3} = \frac{x - 8}{2x + 3}\).
• Next, perform cross multiplication by taking the numerator of the first fraction and multiplying it by the denominator of the second fraction, and vice versa.
• This leads to: \((5x + 2) \times (2x + 3) = (x - 8) \times (10x - 3)\).

By cross-multiplying, you transform the equation into a format without fractions, making it easier to solve.

Equation Expansion

Once you've done cross multiplication, the next step is to expand the terms. Expanding involves distributing each term in a bracket and applying the distributive property, \(a(b + c) = ab + ac\).
This helps in simplifying the problem to a polynomial equation form where you can easily spot like terms.

For our equation:

• On the left side: from \((5x + 2)(2x + 3)\), use the distributive property to get: \(5x \times 2x + 5x \times 3 + 2 \times 2x + 2 \times 3 = 10x^2 + 15x + 4x + 6\).
• Combine similar terms to make it simpler: \(10x^2 + 19x + 6\).
• For the right side: from \((x - 8)(10x - 3)\), similarly distribute the terms to get: \(x \times 10x + x \times -3 - 8 \times 10x - 8 \times -3 = 10x^2 - 3x - 80x + 24\).
• Simplify to: \(10x^2 - 83x + 24\).

Expanding the equation provides a clear pathway to rearranging and simplifying further.

Solving Linear Equations

After expanding, combine like terms on both sides and rearrange to form a standard equation. The goal is to isolate the variable. Here, you'll focus on collecting all terms involving \(x\) on one side and constants on the other.

For instance:

• Begin with: \(10x^2 + 19x + 6 = 10x^2 - 83x + 24\).
• Subtract \(10x^2\) from both sides to eliminate the quadratic term: \(19x + 6 = -83x + 24\).
• Bring all \(x\) terms together by adding \(83x\) to both sides: \(19x + 83x + 6 = 24\).
• This yields \(102x + 6 = 24\).
• Subtract 6 from both sides to start isolating the variable: \(102x = 18\).

Solution progresses by systematically isolating the variable using basic arithmetic operations.

Fraction Simplification

Arriving at a solution like \(x = \frac{18}{102}\), the final step is simplifying fractions. Simplification involves reducing the fraction to its lowest terms by finding the greatest common divisor (GCD).
By simplifying, you express the solution in the most understandable form.

To simplify \(\frac{18}{102}\):

• Find the GCD of 18 and 102. Once determined, divide both the numerator and the denominator by their GCD. In this case, the GCD is 6.
• Divide: \(\frac{18 \div 6}{102 \div 6} = \frac{3}{17}\).

Thus, \(x = \frac{1}{17}\) is the simplest form. Simplification is crucial for clear and accurate results.