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Chapter 3: Problem 26

Solve and check each equation. $$-8 b=-24$$

Short Answer

Expert verified
To solve and check the equation \(-8b = -24\), we first divide both sides by -8, giving us \(b = 3\). Upon checking by substituting b back into the original equation, we confirm our solution is correct. Thus, \(b = 3\).

Step by step solution

01

Identify the operation to isolate the variable.

In this equation, we see that the variable b is being multiplied by -8. To isolate b, we need to perform the inverse of this operation, which is to divide both sides of the equation by -8.
02

Divide both sides by -8.

We'll now divide both sides of the equation by -8: \(-8b=-24 \implies \frac{-8b}{-8} = \frac{-24}{-8}\)
03

Simplify.

After dividing both sides by -8, we can simplify the equation: \(b = 3\)
04

Check the solution.

Now we will check our solution by plugging the value of b back into the original equation: \(-8b = -24 \implies -8 \times 3 = -24\) This simplifies to: \(-24 = -24\) Since both sides are equal, our solution is correct. The answer is \(b = 3\).

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Key Concepts

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

Inverse Operations
Inverse operations are key when solving equations because they allow us to reverse operations to solve for the variable. Inverse operations "undo" each other, just like addition undoes subtraction and multiplication undoes division. In the equation \(-8b=-24\), the variable \(b\) is multiplied by \(-8\). To "undo" this multiplication, we use the inverse operation, division, by \(-8\).
Remember, what you do to one side, you must do to the other side to keep the equation balanced. By dividing both sides by \(-8\), we isolate \(b\), which simplifies our solving process. Knowing which inverse operation to use comes from identifying the operation happening to the variable.
Isolating Variables
Isolating variables is a crucial part of solving equations. It requires getting the variable alone on one side of the equation to find its value. Let's take our example: \(-8b = -24\).
To isolate \(b\), we need it alone on one side. We identified the operation, which was multiplication by \(-8\). By applying the inverse operation of dividing both sides by \(-8\), we achieved isolation.
  • This process simplifies to \(b = \frac{-24}{-8}\).
  • After simplification, \(b = 3\).
Isolating variables often involves a series of strategic steps to "peel away" other components from the variable. It's about using operations strategically to untangle the equation.
Checking Solutions
After finding the solution to an equation, it is important to check that it's correct. This step confirms your work is accurate and the solution satisfies the original equation.
In the example, we found \(b = 3\), but to ensure accuracy, we substitute \(b\) back into the original equation: \(-8b = -24\).
Performing the check means substituting as shown: \(-8 \times 3 = -24\), which simplifies to \(-24 = -24\). Since both sides of the equation match, our solution is correct.
  • Checking helps catch mistakes early.
  • It assures the solution meets the equation's requirements.
By verifying solutions, confidence in your answer's authenticity improves greatly.
Simplifying Equations
Simplifying equations means making them easier to solve by reducing complexity. This includes combining like terms and reducing fractions when necessary. In our example, after dividing by \(-8\), the equation \(b = \frac{-24}{-8}\) simplifies to \(b = 3\).
Simplifying often occurs during or after isolating the variable. It's about 'cleaning up' the math to make the equation and solution as clear and straightforward as possible. It makes the solution not only easier to find but also easier to confirm when checking your work.
  • Reduction helps avoid common mistakes.
  • It ensures each step is clear and correct.
Simplified equations are less prone to errors and maintain the integrity of the original problem.

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