Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$(5 x-7)(2 x+1)-10 x(x-4)=0$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{7}{31}\).

Step 1: Expand the Expressions

First, we need to expand the two expressions in the given equation. Expand \((5x-7)(2x+1)\) and \(-10x(x-4)\).

Step 2: Apply the Distributive Property

Apply the distributive property to expand \((5x-7)(2x+1)\). Multiply each term in the first parenthesis by each term in the second:\[5x \cdot 2x + 5x \cdot 1 - 7 \cdot 2x - 7 \cdot 1 = 10x^2 + 5x - 14x - 7.\]

Step 3: Simplify the First Expansion

Combine like terms in the expanded expression:\[10x^2 + 5x - 14x - 7 = 10x^2 - 9x - 7.\]

Step 4: Expand and Simplify the Second Term

Expand and simplify \(-10x(x-4)\) using the distributive property:\[-10x \cdot x + 10x \cdot 4 = -10x^2 + 40x.\]

Step 5: Combine Both Parts and Simplify

Now, combine the simplified expressions from Steps 3 and 4:\[(10x^2 - 9x - 7) - (10x^2 - 40x) = 0.\]Simplify by combining like terms:\[10x^2 - 9x - 7 - 10x^2 + 40x = 31x - 7.\]Thus, the equation simplifies to:\[31x - 7 = 0.\]

Step 6: Solve for x

To find \(x\), solve the equation:\[31x - 7 = 0.\]Add 7 to both sides:\[31x = 7.\]Divide both sides by 31:\[x = \frac{7}{31}.\]

Key Concepts

Equation solving

Solving equations is like finding the missing piece of a puzzle. An equation tells us that two expressions are equal. To solve it, we need to manipulate the equation to isolate the unknown variable, usually referred to as \(x\). This process involves a series of logical steps that simplify the equation until the variable stands alone on one side of the equation.
For instance, take this step-by-step approach:

• Identify the operations used in the equation, such as addition, subtraction, multiplication, and division.
• Work with inverse operations to reverse these actions. This helps to isolate the variable.
• Maintain balance by performing any operation on both sides of the equation.

In the given exercise, once we've simplified the expressions, we arrive at \(31x - 7 = 0\). To solve this, we add 7 to both sides and then divide by 31, isolating \(x\) and finding its value to be \(\frac{7}{31}\). This method is systematic and ensures you arrive at the correct solution.

Distributive property

The distributive property is a fundamental principle in algebra that helps in breaking down expressions with parentheses. When you see an expression like \((a + b)(c + d)\), the distributive property lets us multiply each term in the first parenthesis by each term in the second parenthesis.
Here's how it works:

• Multiply each term inside one parenthesis by every term in the other parenthesis. This can be remembered as the distributive law: \(a(b + c) = ab + ac\).
• Ensure every term is multiplied correctly to avoid mistakes in the simplification process.

In the exercise, applying the distributive property to \((5x-7)(2x+1)\) involves multiplying terms as shown: \(5x \cdot 2x + 5x \cdot 1 - 7 \cdot 2x - 7 \cdot 1 = 10x^2 + 5x - 14x - 7\). This step is key to expanding and simplifying expressions correctly.

Simplifying expressions

Simplifying expressions is essential to make an equation easier to solve. It involves combining like terms and reducing expressions to their simplest form. Like terms are terms that have the same variables raised to the same power.
For example:

• In the expression \(10x^2 + 5x - 14x - 7\), \(5x\) and \(-14x\) are like terms.
• Combine these like terms to simplify: \(5x - 14x = -9x\), leading to \(10x^2 - 9x - 7\).

Additionally, in the step for \(-10x(x-4)\), simplifying requires distributing and combining any like terms: \(-10x \cdot x = -10x^2\) and \(10x \cdot 4 = 40x\), simplifying to \(-10x^2 + 40x\).
Effective simplification ensures the equation is manageable and ready to be solved, as seen when we arrive at \(31x - 7 = 0\) after combining all parts of the expanded expressions.