Problem 30 - Prealgebra

Question

Solve each equation. Check your solution. $$4=-\frac{1}{8} q$$

Step-by-Step Solution

Answer

The solution is \( q = -32 \).

Step 1: Isolate the variable

Our goal is to solve the equation \( 4 = -\frac{1}{8} q \) for \( q \). To do this, we need to isolate \( q \) on one side of the equation. Start by eliminating the fraction in front of \( q \) by multiplying both sides by \(-8\), the reciprocal of \(-\frac{1}{8}\). This gives us: \[-8 \times 4 = \left(-8 \times -\frac{1}{8}\right) q \] Simplifying this, we get: \[-32 = q\]

Step 2: Verify the solution

Verify that \( q = -32 \) is correct by substituting \( -32 \) back into the original equation:\[ 4 = -\frac{1}{8} \times (-32) \] Calculate the right side: \[-\frac{1}{8} \times (-32) = 4\] Since the left side equals the right side, our solution is verified.

Key Concepts

Understanding Variables

Variables are a fundamental concept in algebra. They are symbols, often letters, that represent unknown values. In an equation, such as \( 4 = -\frac{1}{8} q \), the letter \( q \) is a variable. The goal in solving such equations is to find the value of the variable that makes the equation true.

• Variables can change depending on the context and conditions given in the problem.
• They allow us to construct equations which can be solved to find desired quantities.

In this exercise, the variable \( q \) is initially in a complex form due to the fraction involved. By isolating the variable, we strive to express \( q \) in terms of a simpler, more explicit value.

Checking Solutions

After finding a potential solution for an equation, it's crucial to verify its correctness. In this problem, we identified \( q = -32 \) as a solution. Checking this entails substituting our result back into the original equation to see if it holds true.
Substituting \( q = -32 \) back, we do:

• Calculate \( -\frac{1}{8} \times (-32) \).
• The result should equal 4, which confirms that the variable value is correct.

If both sides of the original equation balance with our solution plugged in, then we can be confident that our solution is accurate. This step helps prevent errors or miscalculations, ensuring reliable results.

Working with Fractions

Fractions can be tricky, especially in solving equations. In this exercise, \( -\frac{1}{8} \) is the coefficient of \( q \). Solving equations involving fractions often requires eliminating the fraction.
To eliminate a fraction:

• Multiply both sides of the equation by the reciprocal of the fraction.
• This process transforms the equation into a simpler, easier-to-solve form.

Here, multiplying by \(-8\), the reciprocal of \(-\frac{1}{8}\), helped us cancel out the fraction, effectively isolating the variable \( q \). This approach streamlines the solution process, making the math more straightforward.

Mastering Algebraic Manipulation

Algebraic manipulation is the technique of rearranging and simplifying equations to find solutions. It involves performing operations like addition, subtraction, multiplication, or division on both sides of an equation to maintain its balance. For example, solving \( 4 = -\frac{1}{8} q \) required:

• Multiplying both sides by \(-8\) to simplify the equation.
• Ensuring that each step maintains the equation's equality.

The efficiency of algebraic manipulation lies in its systematic approach. It allows us to solve complex equations systematically and accurately. Mastery of these techniques is foundational for tackling more intricate algebraic problems.