Problem 54 - Algebra 1: Concepts and Skills

Question

MULTIPLE CHOICE Which operation would you use to solve \(\frac{1}{4}=-8 x ?\) A. Divide by 4. B. Divide by \(-8\) C. Multiply by \(-8 .\) D. Multiply by 4

Step-by-Step Solution

Answer

The correct choice is B. Divide by \(-8\).

Step 1: Understanding the Problem

We are given the equation \(\frac{1}{4} = -8x\). We need to solve for x. The task is to decide which operation to apply to both sides of the equation to isolate x.

Step 2: Identify the Proper Operation

Observing the equation, we see an '-8' being multiplied by 'x'. To cancel out multiplication, the reciprocal operation is division. Therefore, we should divide by '-8'.

Step 3: Apply the Operation to the Equation

We divide both sides of the equation by -8. This gives us \(x = -\frac{1}{4} \div -8\). By simplifying, we find \(x = \frac{1}{32}\).

Key Concepts

Algebraic Operations

Algebraic operations are fundamental tools used to manipulate and solve equations. In the exercise, we are tasked with solving for a variable, "x". This requires understanding the operations present in our equation. Since \(-8x\) suggests multiplication, recognizing this operation is crucial for deciding how to proceed.

These operations include:

• Addition - Combining values; the opposite operation is subtraction.
• Subtraction - Removing value; its opposite is addition.
• Multiplication - Repeated addition; its opposite is division.
• Division - Splitting into parts; countered by multiplication.

Understanding these operations helps identify the correct steps needed to simplify or transform an equation into a solvable state. In more complex scenarios, multiple operations might be used, requiring careful step-by-step execution to maintain balance in the equation.

Multiplication and Division in Equations

When solving equations, multiplication and division are often used as inverse operations to isolate variables. In our exercise, \(\frac{1}{4} = -8x\), "-8" is multiplying by "x". To undo this multiplication, division is applied.

Here's how it works:

• Multiplication - Increases the value of a variable; move to the opposite through division.
• Division - Decreases or distributes a value; countered by multiplication.

By dividing both sides of the equation by the coefficient of "x", which is "-8", we effectively remove the multiplication by "-8". This simplifies solving the equation, as each side of the equation still remains balanced.

Isolating Variables

One of the main goals when solving any equation is isolating the variable. This means getting the variable by itself on one side of the equation. Why is this important? Because it reveals the solution to the equation. Steps to isolate include reversing operations following the order of operations in reverse (PEMDAS/BODMAS) where needed.

To isolate "x" in the exercise \(\frac{1}{4} = -8x\):

• Identify the operation - Here, it is multiplication by "-8".
• Perform the inverse operation - Divide by "-8" on both sides.
• Simplify - Calculate the result to get "x" on its own.

By following these steps, \(x = \frac{1}{32}\) is revealed, showing the value that satisfies the original equation. Isolating variables uses algebraic insight to transparently solve equations, turning complex problems into simple solutions.