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Chapter 1: Problem 21

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

Short Answer

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

Step by step solution

01

Distribute Constants

First, distribute the constants 2 on the left side and 3 on the right side of the equation. This results in: \[ 2 - 2x = 3 + 6x + 5 \]
02

Simplify Equation

Combine like terms on the right side of the equation. Add 3 and 5:\[ 2 - 2x = 8 + 6x \]
03

Isolate Variable Term

Move the \( -2x \) to the right side by adding \( 2x \) to both sides:\[ 2 = 8 + 8x \]
04

Isolate Constant Term

Subtract 8 from both sides to move the constant from the right side to the left:\[ 2 - 8 = 8x \]This simplifies to:\[ -6 = 8x \]
05

Solve for the Variable

To isolate \( x \), divide both sides of the equation by 8:\[ \frac{-6}{8} = x \]Simplify \( \frac{-6}{8} \) to \( \frac{-3}{4} \): \[ x = \frac{-3}{4} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation is usually in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. The process can be straightforward if approached methodically, typically involving undoing operations with inverse operations. Keep a few basics in mind:
  • Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced, much like a seesaw.
  • You focus on getting the variable by itself on one side of the equation by applying opposite operations. For instance, if a variable is subtracted, you add to both sides, and if it is multiplied, you divide both sides.
As you get comfortable with the idea of maintaining balance, solving equations will become a smoother process.
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that helps you handle expressions where a number multiplies a sum or difference inside parentheses. It states that: \[ a(b + c) = ab + ac \]This property allows you to "distribute" the multiplication over addition or subtraction inside the parentheses. In the given problem, the expression \( 2(1-x) \) means you multiply \( 2 \) by both \( 1 \) and \( -x \), giving you \( 2 - 2x \).
  • This step simplifies expressions and helps in further isolating the variable.
  • It's an essential skill in algebra, making complicated expressions easier to solve by removing parentheses early in your work.
Once you get the hang of this, you'll find that applying the distributive property becomes second nature in solving linear equations.
Isolation of Variables
The main goal in solving equations is to get the variable by itself. Here's a more detailed look at how to isolate variables effectively:
  • After expanding any terms using the distributive property, combine like terms on each side of the equation to simplify. This means adding similar terms together, such as constants with constants and variable terms with each other.
  • Once simplified, move the variable terms to one side by either adding or subtracting them from both sides. This helps you start isolating the variable. For example, if you have \( 2 - 2x = 8 + 6x \), add \( 2x \) to both sides to start isolating \( x \).
  • Next, tackle the constants. Move any constants on the same side as the variable to the opposite side, continuing to simplify the equation towards isolating \( x \).
Each step brings you closer to solving the equation. Slow and steady isolation makes the process less daunting.
Simplifying Fractions
Once you've isolated the variable, you may need to simplify fractions to provide the cleanest answer. Simplifying fractions involves reducing the fraction to its lowest terms by dividing both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
  • In the solution \( x = \frac{-6}{8} \), the GCD of \( 6 \) and \( 8 \) is \( 2 \). So, you divide both \( -6 \) and \( 8 \) by \( 2 \) to simplify the fraction to \( \frac{-3}{4} \).
  • Always check if the fraction can be simplified further. Simplifying fractions helps make your final answer clear, concise, and correct.
Being diligent in this step ensures that your solutions are not only accurate but also elegantly expressed.

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Most popular questions from this chapter

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