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Chapter 2: Problem 8

Solve and check. Label any contradictions or identities. $$3 x+6=18$$

Short Answer

Expert verified
x = 4

Step by step solution

01

Isolate the variable term

Start by subtracting 6 from both sides of the equation to isolate the term with the variable. This gives us:\[3x + 6 - 6 = 18 - 6\]which simplifies to:\[3x = 12\]
02

Solve for x

Next, divide both sides of the equation by 3 to solve for x. This gives us:\[\frac{3x}{3} = \frac{12}{3}\]which simplifies to:\[x = 4\]
03

Check the solution

Substitute x = 4 back into the original equation to check the solution. The original equation is:\[3x + 6 = 18\]Substitute x = 4:\[3(4) + 6 = 18\] This simplifies to:\[12 + 6 = 18\]which is true. Therefore, x = 4 is the correct solution.
04

Identify contradictions or identities

There are no contradictions or identities in this solution. The solution x = 4 is valid and consistent.

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Key Concepts

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

Isolating the Variable
Isolating the variable is the first step in solving linear equations. Our goal here is to get the variable, typically denoted as x, by itself on one side of the equation.
We start with the equation: \(3x + 6 = 18\).
The first step towards isolating x is to eliminate the constant term (6 in this case) that's on the same side as the variable term (3x).
To do this, we perform the same operation (subtraction in this case) on both sides of the equation. We subtract 6 from both sides:
\[3x + 6 - 6 = 18 - 6\] This simplifies to:
\[3x = 12\]
Now, with the term involving x isolated, we proceed to the next step of solving for x.
Checking the Solution
Checking the solution is crucial after solving the equation. This ensures that our solution is correct and satisfies the original equation. Once we have found \(x = 4\), we substitute it back into the original equation:
\(3x + 6 = 18\)
Substituting \(x = 4\) gives us:
\[3(4) + 6 = 18\]
Simplifying further:
\[12 + 6 = 18\]
Since both sides of the equation are equal, \(x = 4\) is indeed the correct solution. This step is important because it confirms that our work is accurate. We always recommend doing this check to avoid mistakes.
No Contradictions
A contradiction in a mathematical solution means that there is no possible value that satisfies the original equation.
This can happen in some cases, but it did not occur in our example. An identity, on the other hand, is a situation where any value for the variable satisfies the equation.
In our example, neither a contradiction nor an identity was present. The equation \(3x + 6 = 18\) had a clear and single solution, \(x = 4\).
It鈥檚 important to check for contradictions or identities to ensure that the solution is valid and consistent.
This step helps in identifying any potential errors or special cases.

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