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Chapter 0: Problem 20

Solve for the indicated variable. $$5-(2 x-3)=7-(3 x+5)$$

Short Answer

Expert verified
Expand, combine like terms, solve for \( x \): \( x = -4 \).

Step by step solution

01

Expand both sides

First, we need to expand the terms inside the parentheses on both sides of the equation. This will give us: \[ 5 - 2x + 3 = 7 - 3x - 5 \]

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Key Concepts

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

Equation Solving
Solving a linear equation involves finding the value of the variable that makes the equation true. In any given equation, the expressions on either side of the equals sign must be equal in value. For example, consider the equation we are solving:
  • Equation: \( 5 - (2x - 3) = 7 - (3x + 5) \).
Breaking down the expression and simplifying both sides helps reveal the variable's value. The goal is to perform basic arithmetic operations such as addition, subtraction, multiplication, or division to isolate the variable.
You can think of equation solving as a balanced scale. The goal is to keep the equation balanced while you simplify both sides until you zero in on the variable's value.
Variable Isolation
One crucial step in solving equations is isolating the variable. The objective is to have the variable on one side of the equation and numerical values on the other.Here's a simple guide to variable isolation:
  • Add or subtract terms on both sides to remove numbers from the side with your variable.
  • Divide or multiply to simplify the coefficient, which is the number multiplying the variable.
By isolating the variable in our problem:
  1. The equation after expansion is \( 5 - 2x + 3 = 7 - 3x - 5 \).
  2. Combine like terms: \( 8 - 2x = 2 - 3x \).
  3. Add \( 3x \) to both sides: \( 8 + x = 2 \).
  4. Subtract \( 8 \) from both sides: \( x = 2 - 8 \).
  5. Solution simplifies to: \( x = -6 \).
Isolating the variable allows for precise solutions and is the key operation for solving equations.
Algebraic Expression
Algebraic expressions are combinations of numbers, variables, and operators that represent a specific value. In an equation, understanding and manipulating these expressions are crucial to finding the variable's value. Let's look closer at our example:The expressions in the initial equation need simplification, especially within the parentheses. This is a typical step in working with algebraic expressions. Breaking them down involves distributive properties and combining like terms:
  • Expression inside parenthesis is essential to shift to basic terms.
  • Simplify expressions: removing parenthesis in \( 5 - (2x - 3) \) gives \( 5 - 2x + 3 \).
  • Look for and combine like terms: consolidate constants and terms with the same variable.
Algebraic expressions form the foundation of equations like building blocks. Mastering simplification and manipulation skills will greatly enhance your algebra capabilities.

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