Problem 86 - Algebra 1

Question

Solve the equation. Check the result. (Review 3.4 ) $$-6(q+22)=5 q$$

Step-by-Step Solution

Answer

The solution to the equation is \( q = -12 \)

Step 1: Distribute -6 Over the Parenthesis

Start by distributing the -6 over \( q+22 \) to get \( -6q - 132 = 5q \)

Step 2: Rearrange Terms

Next, add \( 6q \) to both sides of the equation to collect all \( q \) terms on one side, this leads to: \( -132 = 5q + 6q \)

Step 3: Simplify Terms

On the right side, combine like terms (5q + 6q) to get \( -132 = 11q \)

Step 4: Solve For q

Finally, divide by 11 on both sides to isolate \( q \). This gives: \( q = -132/11 \)

Step 5: Check the Solution

To check if our solution is correct, we substitute \( q = -12 \) back into the original equation: \( -6(-12+22) = 5(-12) \). Both sides evaluate to be -120, thus confirming \( q = -12 \) as the correct solution

Key Concepts

Distributive Property

Understanding the distributive property is crucial to solving linear equations. It's a mathematical principle that allows us to remove parentheses by distributing a factor across each term inside the parenthesis. For example, with the equation \( -6(q+22) = 5q \), you apply the distributive property by multiplying \( -6 \) with both \( q \) and \( 22 \), leading to \( -6q - 132 \). This step is foundational to simplifying the equation and moving towards a solvable format.

Use the distributive property every time you notice parentheses combined with a numerical factor. This action simplifies your equation and prepares it for subsequent steps, like combining like terms and isolating the variable.

Combining Like Terms

Once you've distributed your factors, the next step in resolving a linear equation is to combine like terms. Like terms are mathematical expressions that have the same variable raised to the same power. In our equation, \( 5q + 6q \) are like terms because they both contain the variable \( q \) raised to the first power.

By combining \( 5q \) and \( 6q \) together, we get \( 11q \), which significantly simplifies the equation. This step is critical because it consolidates the variable terms into a single term, making it easier to isolate the variable in subsequent steps.

Equation Checking

Verifying your solution by checking the equation is a good practice to ensure you've arrived at the correct answer. When solving \( -6(q+22)=5q \) and finding that \( q = -12 \), it's important to plug \( -12 \) back into the original equation. This means you replace \( q \) with \( -12 \) and perform the calculations:

\( -6(-12+22) = 5(-12) \)
which simplifies to \( -6(10) = -60 \), and \( 5(-12) = -60 \), confirming both sides equal \( -60 \) (not \( -120 \) as originally described in the solution).

This proof reinforces your confidence in the validity of the solution. Always remember to check your answers to avoid simple errors.

Isolating Variables

The goal of solving linear equations is to find the value of an unknown variable. Once you've combined like terms and simplified the equation, you isolate the variable, which entails getting the variable on one side of the equation, and all numbers on the other. From our combined like terms \( -132 = 11q \), we isolate \( q \) by dividing both sides by \( 11 \) resulting in \( q = -12 \).

\( q = \frac{-132}{11} \) is the isolated variable, providing us the solution to the equation. Isolating the variable is a fundamental aspect of solving equations and mastering this technique is essential for algebraic success.