Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{17}\).

Step 1: Cross multiply

To eliminate the fractions, we will cross multiply the terms. This means we multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction.\((5x + 2)(2x + 3) = (10x - 3)(x - 8)\).

Step 2: Expand both sides of the equation

We expand both sides of the equation by applying the distributive property. For the left side, we distribute \(5x+2\) across \(2x+3\), and for the right side, distribute \(10x-3\) across \(x-8\):Left side: \(5x \cdot 2x + 5x \cdot 3 + 2 \cdot 2x + 2 \cdot 3\) = \(10x^2 + 15x + 4x + 6\).Right side: \(10x \cdot x - 10x \cdot 8 - 3 \cdot x + 3 \cdot 8\) = \(10x^2 - 80x - 3x + 24\).

Step 3: Simplify both sides

Combine like terms for each side:Left side becomes: \(10x^2 + 19x + 6\).Right side becomes: \(10x^2 - 83x + 24\).

Step 4: Move all terms to one side

Subtract the terms on the right side from the left side to set the equation to zero:\(10x^2 + 19x + 6 - (10x^2 - 83x + 24) = 0\).Simplify this to get: \(0x^2 + 102x - 18 = 0\), which simplifies to \(102x = 18\).

Step 5: Solve for x

Divide both sides by 102 to solve for \(x\):\(x = \frac{18}{102}\).Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6:\(x = \frac{3}{17}\).

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations involving fractions, particularly rational equations. This method involves the cross product of the terms of two ratios set equal to each other.
In the given equation \(\frac{5x+2}{10x-3} = \frac{x-8}{2x+3}\), cross multiplication allows you to eliminate the denominators by multiplying across the diagonal.

• Multiply the numerator of the first fraction by the denominator of the second fraction: \((5x + 2)(2x + 3)\).
• Multiply the denominator of the first fraction by the numerator of the second fraction: \((10x - 3)(x - 8)\).

Set these products equal: \((5x + 2)(2x + 3) = (10x - 3)(x - 8)\).
This ensures that the equation is balanced. By building a cross-multiplication bridge, students can seamlessly transition into solving rational equations without fractions.

Distributive Property

The distributive property is a fundamental algebraic principle used to expand expressions like those in the cross-multiplied form. It allows you to distribute or "spread" a multiplication operation over addition or subtraction inside parentheses.
When applying this to our equation, consider:

• \((5x + 2)(2x + 3)\): Multiply each term in the first polynomial by each term in the second polynomial.
• \((10x - 3)(x - 8)\): Apply the same distribution.

For the left side:

• \(5x \cdot 2x\) and \(5x \cdot 3\).
• \(2 \cdot 2x\) and \(2 \cdot 3\).

Combined: \(10x^2 + 15x + 4x + 6\).
On the right side:

• \(10x \cdot x\) and \(10x \cdot -8\).
• \(-3 \cdot x\) and \(-3 \cdot -8\).

Combined: \(10x^2 - 80x - 3x + 24\).
The distributive property simplifies the equation into a form suitable for further algebraic manipulation.

Simplifying Equations

Simplifying equations involves combining like terms to reduce them to their simplest form, which makes solving them easier. After distributing in the previous step, the next step is to simplify each side of the equation.
Let's look at each side:

• Left side: Combine the \(x\) terms into \(19x\) as \(15x + 4x\), resulting in \(10x^2 + 19x + 6\).
• Right side: Combine the \(x\) terms into \(-83x\) as \(-80x - 3x\), resulting in \(10x^2 - 83x + 24\).

Once simplified, transfer everything to one side to compare all terms against zero:
\(10x^2 + 19x + 6 - (10x^2 - 83x + 24) = 0\).
Simplification helps clear the way for the next step, leading smoothly into solving.

Solving for x

With a streamlined equation, solving for \(x\) becomes straightforward. The equation has been simplified to \(102x - 18 = 0\).
The task now is to isolate \(x\).

• Add all like terms to one side and constants to the other.
• Divide both sides by the coefficient of \(x\), which is 102, to solve for \(x\).

Thus, \(102x = 18\) turns into \(x = \frac{18}{102}\).
It’s always important to simplify the final fraction. Here, dividing both the numerator and the denominator by their greatest common divisor (6), we get \(x = \frac{3}{17}\).
Finding \(x\) simplifies the equation's complexity, rendering a solution that showcases one's understanding of rational equations.