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Chapter 1: Problem 2

Solve and check each linear equation. $$6 x-3=63$$

Short Answer

Expert verified
The solution is \(x=11\), and checking the equation confirms that this is the correct solution.

Step by step solution

01

Isolate the Variable Term

To isolate the term containing the variable \(x\) on one side of the equation, add \(3\) to both sides of the equation. This will cancel the \(-3\) on the left side of the equation. Therefore, the equation becomes: \[6x - 3 + 3 = 63 + 3\] Which simplifies to: \[6x = 66\]
02

Solve for the Variable

To solve for \(x\), divide each side of the equation by \(6\) as the equation is \(6x = 66\). This will give: \[x = 66/6 = 11\]
03

Checking the Solution

Substitute \(11\) for \(x\) in the original equation \(6x - 3 = 63\), then simplify the left-hand side: \[6(11) - 3 = 66 - 3 = 63\] Which is the same as the right-hand side of the original equation, thus confirming \(x = 11\) is the correct solution.

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Key Concepts

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

Solving Equations
Solving equations is a fundamental skill in algebra that involves finding the unknown value that balances the equation. In the problem given, we need to solve the linear equation \(6x - 3 = 63\). Calculating the solution means performing operations to both sides of the equation to keep them equal.
  • Start by simplifying both sides of the equation if needed. This means combining like terms or eliminating fractions.
  • Our main goal is to have the variable \(x\) on one side and the constants on the other.
  • This usually involves using addition, subtraction, multiplication, or division.
Keeping the equation balanced is critical. This means whatever operation you perform on one side, you must also perform on the other side. This ensures that the equation remains valid until you can isolate the variable.
Isolating Variables
Isolating variables is an essential step that enables us to solve equations. Think of it like peeling layers of an onion until only the core remains, which in this case, is the variable. In our exercise, we aim to isolate \(x\). Starting with the equation \(6x - 3 = 63\), we'll isolate the variable by eliminating the constant term next to it.
  • Start by adding \(3\) to both sides: \(6x - 3 + 3 = 63 + 3\).
  • This simplifies to \(6x = 66\), effectively removing the constant term from the left side.
  • Next, divide both sides by \(6\) to get \(x\) alone: \(x = 66/6\).
Now, \(x\) is isolated on one side of the equation, and we've found its value: \(x = 11\). Remember, the key is to perform inverse operations that undo what has been done to the variable, leading it to stand by itself.
Checking Solutions
Checking solutions is the final step to confirm that what you found is the correct answer. This part helps verify that no errors were made during the process of solving the equation. To check the solution of \(x = 11\) for the equation \(6x - 3 = 63\):
  • Substitute \(11\) back into the equation for \(x\): \(6(11) - 3\).
  • Simplify the expression: \(66 - 3 = 63\).
  • Observe if the result on the left side equals the original right side.
Since both sides of the equation are equal (63 = 63) after substituting the solution, this confirms that our solution \(x = 11\) is valid. Checking solutions acts as a safeguard to ensure accuracy, providing peace of mind that the solution is indeed correct.

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