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Chapter 3: Problem 10

Solve each equation. First combine any like terms on each side of the equation. \(-3 x=11-2\)

Short Answer

Expert verified
The solution is \(x = -3\).

Step by step solution

01

Simplify the Right Side of the Equation

First, we need to simplify the expression on the right side of the equation. The term we have is \(11 - 2\), which simplifies to \(9\). So the equation becomes \(-3x = 9\).
02

Isolate the Variable

To solve for \(x\), we need to isolate it on one side of the equation. Since \(x\) is being multiplied by \(-3\), we do this by dividing both sides of the equation by \(-3\). This gives us \(x = \frac{9}{-3}\).
03

Simplify the Result

Now simplify the fraction on the right side of the equation. Dividing \(9\) by \(-3\) gives \(-3\). So, \(x = -3\).

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Key Concepts

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

Combining Like Terms
Combining like terms is a fundamental concept when solving linear equations. It involves merging terms that have the same variable or constant to simplify equations. For example, consider the expression evenly distributed in an equation like:
  • \( 2x + 3x = 5x \)
  • \( 7 - 2 = 5 \)
In these cases, terms with the same variable \(x\) or identical numerical terms can be added together to combine them. This process will not only make the equations simpler but also prepare you for the next steps in solving. Look for constants or terms with similar variables and merge them into a single term. This reduces clutter and gives a clearer picture of the equation.
Simplifying Equations
Simplifying equations is the act of making an equation easier to understand and solve. It involves performing operations like combining like terms, as well as calculating basic arithmetic on both sides. Let's explore this concept more deeply. Take the equation:
  • \(-3x = 11 - 2\)
Start by calculating expressions that can be simplified, such as \(11 - 2\), which becomes \(9\). Thus, the equation changes to \(-3x = 9\).
Simplification aims to transform the equation into a format that is easier to work with, allowing us to clearly see what operations we need to perform next. It often involves performing arithmetic operations and arranging terms for easier manipulation.
Isolating the Variable
Isolating the variable is crucial for finding the solution of an equation. It involves rearranging the equation so that the variable we are solving for is by itself on one side. This enables us to see what value the variable represents. For example, in the equation:
  • \(-3x = 9\)
To isolate \(x\), divide both sides by \(-3\). Performing this operation gives:
  • \(x = \frac{9}{-3}\)
After further simplification, you find that \(x = -3\). The goal is to have the variable appear by itself, making it easy to determine its value. Make sure to perform the same operation on both sides of the equation to maintain equality. This step is pivotal in moving from an abstract expression to a concrete solution.

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