Problem 18 - Prealgebra

Question

Solve each equation. Check your solution. $$11=\frac{b}{-3}$$

Step-by-Step Solution

Answer

The solution is \(b = -33\). The check confirms it's correct.

Step 1: Multiply Both Sides by -3

To solve for \(b\) in the equation \(11 = \frac{b}{-3}\), we start by eliminating the fraction. Multiply both sides of the equation by \(-3\) to isolate \(b\) on one side. The equation becomes:\[11 imes (-3) = \frac{b}{-3} imes (-3)\]This simplifies to:\[-33 = b\]Now \(b\) is isolated.

Step 2: Check the Solution

To ensure our solution is correct, substitute the value of \(b\) back into the original equation and verify if both sides are equal. The original equation is:\[11 = \frac{b}{-3}\]Substituting \(b = -33\) gives:\[11 = \frac{-33}{-3}\]This simplifies to:\[11 = 11\]Since both sides are equal, the solution \(b = -33\) is correct.

Key Concepts

Checking Solutions

Once you've proposed a solution to an equation, it’s vital to check whether it's correct. This ensures accuracy and helps prevent mistakes. To do this, substitute the calculated value back into the original equation and confirm both sides are equal. It's like a mini-test for your solution! For example, if you solved the equation \(11 = \frac{b}{-3}\) and found \(b = -33\), you should plug \(-33\) back into the equation:

• Original Equation: \(11 = \frac{b}{-3}\)
• Substituted Equation: \(11 = \frac{-33}{-3}\)

These steps should show both sides are equal, making sure your solution is correct. This practice not only reinforces confidence but helps you understand solving equations better. Remember, checking solutions is a key step in any math problem-solving process!

Fraction Elimination

Fractions can complicate equations by cluttering the arithmetic. A practical trick to simplify equations is to eliminate fractions early on. You do this by multiplying both sides by the denominator of the fraction, thereby simplifying the equation to a more manageable form.In the equation \(11 = \frac{b}{-3}\), the fraction \(\frac{b}{-3}\) is simplified by multiplying both sides by \(-3\):

• Multiply both sides: \(11 \times (-3) = \frac{b}{-3} \times (-3)\)
• Simplify: \(-33 = b\)

Through this method, the fraction is eliminated, and you're left with a straightforward equation that’s easier to work with. This step is crucial as it leads directly to the solution.

Equation Verification

Verifying an equation after solving it is like giving your solution a seal of confidence. This step involves double-checking your mathematical manipulations to ensure no mistakes were made.After finding a solution, revisit the problem to review each step:

• Recalculate any arithmetic operations completed during solving.
• Check each transformation, ensuring no rules of algebra were violated.

For example, in the problem \(11 = \frac{b}{-3}\), you verified \(b = -33\) by substituting back into the equation. If every action checks out, your solution stands confirmed. Being meticulous in equation verification can highlight careless errors and strengthen your understanding, giving you clarity and certainty in your work.