Problem 23 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(2(1-x)=3(1+2 x)+5\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{-3}{4} \).

Step 1: Expand Both Sides

Start by expanding the expressions on both sides of the equation. We have: \( 2(1-x) \) on the left and \( 3(1+2x) + 5 \) on the right. Expand each:\[2(1-x) = 2 - 2x\]\[3(1+2x) + 5 = 3 + 6x + 5\]Simplifying the right side gives us:\[3 + 6x + 5 = 8 + 6x\]So, the equation becomes: \( 2 - 2x = 8 + 6x \).

Step 2: Get All Variables on One Side

To solve for \( x \), we need to get all terms containing \( x \) on one side of the equation. Add \( 2x \) to both sides:\[2 - 2x + 2x = 8 + 6x + 2x\]This simplifies to:\[2 = 8 + 8x\].

Step 3: Isolate the Variable Term

Subtract 8 from both sides to isolate the term with \( x \):\[2 - 8 = 8x\]Which simplifies to:\[-6 = 8x\].

Step 4: Solve for x

Divide both sides by 8 to solve for \( x \):\[\frac{-6}{8} = x\]Simplify \( \frac{-6}{8} \) to \( \frac{-3}{4} \). Thus, \( x = \frac{-3}{4} \).

Key Concepts

Solving Linear Equations

Linear equations are a fundamental part of algebra, describing a straight-line relationship between variables. To solve them means finding the values for the variables that make the equation true. For example, consider the equation \(2(1-x)=3(1+2x)+5\). The process involves rearranging the equation until the variable is isolated, providing its exact value. This usually includes using arithmetic operations like addition, subtraction, multiplication, and division.

• Identify the variable that needs to be solved, usually represented with \(x\).
• Apply algebraic operations to simplify the equation and isolate the variable.

By effectively solving these equations, students can apply this understanding to a variety of scientific and real-world problems.

Distributive Property

The distributive property is a key technique in algebra, allowing you to expand expressions and simplify equations. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding. For instance, with \(2(1-x)\) and \(3(1+2x)\), apply the distributive property:

• \(2(1-x) = 2 \times 1 - 2 \times x = 2 - 2x\)
• \(3(1+2x) = 3 \times 1 + 3 \times 2x = 3 + 6x\)

Using distributive property simplifies complex equations, making it easier to combine like terms and solve for variables.

Isolating Variables

Isolating the variable is the process of getting the unknown by itself on one side of an equation. This involves moving other terms to the opposite side, using inverse operations. Take the linear equation \(2 - 2x = 8 + 6x\), where \(x\) terms need to be on one side.

• Add \(2x\) to both sides: \(2 - 2x + 2x = 8 + 6x + 2x\).
• Subtract 8 from both sides: \(2 - 8 = 8x\).

By performing these steps systematically, you maintain the balance of the equation while narrowing down the focus to the variable of interest.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form, making it easier to solve. After applying the distributive property, collect like terms and perform basic arithmetic to simplify. For example:

• The equation \(3 + 6x + 5 = 8 + 6x\) simplifies by combining like terms to \(8 + 6x\).

Once simplified, the focus turns to applying known operations to solve for the variable, ensuring a clear, straightforward path to the solution. This step is crucial as it minimizes unnecessary complexity, allowing easier interpretation and solution of the equation.