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Chapter 11: Problem 13

Solve using the addition principle. $$ x+16=-2 $$

Short Answer

Expert verified
x = -18

Step by step solution

01

Understand the Addition Principle

The addition principle states that if you add or subtract the same number from both sides of an equation, the two sides remain equal. This helps in isolating the variable to solve for it.
02

Isolate the Variable

To isolate the variable, subtract 16 from both sides of the equation to cancel out the +16 on the left side: \[ x + 16 - 16 = -2 - 16 \]
03

Simplify Both Sides

Perform the subtraction on both sides of the equation: \[ x = -18 \]

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Key Concepts

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

Algebraic Equations
Algebraic equations are mathematical statements that assert the equality of two expressions. They often include a variable, which is an unknown quantity represented by a letter (usually x, y, or z). In algebraic equations, the goal is to find the value of this variable that makes the equation true. For example, in the problem

x + 16 = -2,

our task is to determine the value of x that turns the left-hand side into the right-hand side. Algebraic equations are foundational in many areas of mathematics and are essential for solving real-world problems.
Variable Isolation
Variable isolation involves manipulating an equation to get the variable by itself on one side of the equation. This process often uses the Addition Principle, which states that you can add or subtract the same number from both sides of an equation without changing its equality. The goal is to simplify the equation step-by-step to isolate the variable. In our example:

x + 16 = -2,

you start by isolating x. You can subtract 16 from both sides to cancel out the +16 on the left side:

x + 16 - 16 = -2 - 16.

This simplifies to:

x = -18.
Solving Linear Equations
Linear equations are equations of the first degree, meaning they have no exponents or powers higher than one. They typically take the form ax + b = c, where a, b, and c are constants. Solving linear equations involves several steps:
  • Identify the variable you need to isolate.
  • Use algebraic principles, like the Addition Principle or Subtraction Principle, to manipulate the equation.
  • Simplify both sides of the equation until the variable is isolated.
Let's look at our example:

x + 16 = -2.

By subtracting 16 from both sides, you isolate x:

x + 16 - 16 = -2 - 16,

which simplifies to:

x = -18.

Therefore,

x = -18 solves the equation, showing how algebraic techniques can simplify and solve for variables.

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