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Chapter 2: Problem 15

Solve each equation and check the result. $$ -5 q-2=23 $$

Short Answer

Expert verified
The solution is \( q = -5 \).

Step by step solution

01

Isolate the Variable

Add 2 to both sides of the equation to move the constant term on the left side to the right side. This gives us \[ -5q = 23 + 2 \]Simplifying the right side:\[ -5q = 25 \]
02

Solve for the Variable

Divide both sides of the equation by -5 to isolate \( q \):\[ q = \frac{25}{-5} \]Simplify the right side:\[ q = -5 \]
03

Check the Solution

Substitute \( q = -5 \) back into the original equation to verify it satisfies the equation:\[-5(-5) - 2 = 23 \]Simplify the left side: \(25 - 2 = 23\).Since both sides of the equation are equal, our solution is verified.

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Key Concepts

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

Isolating the Variable
In solving linear equations, one of the fundamental steps is isolating the variable. When faced with an equation like \[-5q - 2 = 23,\]the first task is to get the term with the variable on one side of the equation by itself.

To achieve this, consider the equation as a balance that you need to keep even. Add the same number to both sides to maintain the balance.
  • Add 2 to both sides, which moves the constant term (-2) on the left side across to the right, leaving you with:
\[-5q = 23 + 2.\]
Simplifying the right side gives \(-5q = 25.\)

Remember, the goal is to isolate \(q.\) So once the constants are managed, you proceed to handle the coefficient of the variable next.
Equation Verification
After finding the solution, it鈥檚 crucial to ensure the answer is correct through equation verification. This is where you plug the value you found back into the original equation.

For instance, having solved that \(q = -5,\) substitute \(q\) back into the original equation:
  • \(-5(-5) - 2 = 23.\)
Simplify the left-hand side:
  • Calculate \(-5 \times -5\) which gives you 25.
  • Then subtract 2 resulting in 23.

Since the left side equals the right side (23 = 23), it confirms that the solution \(q = -5\) is correct. This step reinstates the accuracy of your solution.
Simplifying Algebraic Expressions
Throughout the process of solving equations, you'll often need to simplify expressions. Simplifying reduces an expression down to its simplest form, making equations easier to solve and understand. Here鈥檚 how it works:

Initially, organize like terms and constants to clear the equation of unnecessary complexity. This is what we did after adding 2 to both sides, which resulted in:
  • \(-5q = 25.\)

Next, divide both sides by the coefficient of the variable to completely isolate it. In this case:
  • \(q = \frac{25}{-5}.\)
  • Then, simplify \(q\) to get \(q = -5.\)

Each simplification step transforms complex parts of an equation into simpler, more workable pieces, which is essential in algebra.

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