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Chapter 2: Problem 48

Solve the given equation for \(\mathrm{x}\). \(-5 x-15=45\)

Short Answer

Expert verified
The solution is \( x = -12 \).

Step by step solution

01

Simplify the Equation

Begin by adding 15 to both sides of the equation to isolate the term with the variable on one side. This yields: \[ -5x - 15 + 15 = 45 + 15 \] which simplifies to: \[ -5x = 60 \]
02

Solve for x

Now, divide both sides of the equation by -5 to solve for \( x \). This gives:\[ \frac{-5x}{-5} = \frac{60}{-5} \] which simplifies to: \[ x = -12 \]

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Key Concepts

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

Linear Equations
Linear equations are foundational in algebra and are among the simplest forms of equations you'll come across. They are called "linear" because they graph as straight lines in a two-dimensional space. A linear equation in one variable can typically be written in the form \( ax + b = c \), where:
  • \( a \), \( b \), and \( c \) are constants.
  • \( x \) is the variable you need to solve for.
In this case, the equation \(-5x - 15 = 45\) fits this description, where \( a = -5 \), \( b = -15 \), and \( c = 45 \).
The goal is to find the value of \( x \) that makes the equation true. Linear equations are straightforward once you understand how to manipulate the terms to isolate the variable.
They form the basis for solving more complex equations later on.
Isolation of Variables
When solving an equation, the main aim is to isolate the variable on one side of the equation. It means having the variable by itself so that you can easily determine its value. In the equation \(-5x - 15 = 45\), this process involves a series of steps:
  • Addition or Subtraction: Start by eliminating any constants on the same side as the variable. This usually involves either adding or subtracting terms. For our equation, adding 15 to both sides achieves this, leading to \( -5x = 60 \).
  • Division or Multiplication: Next, use division or multiplication to get the variable alone. Here, dividing both sides by \(-5\) isolates \( x \) and gives the solution \( x = -12 \).
Breaking the equation down in such steps makes it easier to manage and understand. With practice, identifying which terms to move and which operations to use becomes second nature.
Inverse Operations
Inverse operations are key in solving equations efficiently. These are operations that "undo" each other and are crucial for isolating variables. Essentially, when you perform an operation on one side of the equation, you perform the inverse operation to simplify or solve it. Here's how they work:
  • Addition and Subtraction: These are inverse operations. For instance, in the equation \(-5x - 15 = 45\), we added 15 to "undo" the \(-15\), effectively neutralizing it.
  • Multiplication and Division: These also work inversely. In solving \(-5x = 60\), dividing both sides by \(-5\) undoes the multiplication by \(-5\), thus solving for \( x \).
Understanding inverse operations helps you to systematically solve equations, as each operation strategically moves the equation towards the desired format with the variable isolated. This technique is vital for tackling a variety of algebraic problems.

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