Problem 29 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{x}{8}=96 $$

Step-by-Step Solution

Answer

Answer: The value of x in the equation is x = 768.

Step 1: Identify the given equation

The given equation is: $$ \frac{x}{8}=96 $$

Step 2: Isolate x

To isolate x, we need to apply the inverse operation to both sides of the equation. In this case, we need to multiply both sides by 8: $$ \frac{x}{8} \times 8 = 96 \times 8 $$

Step 3: Simplify the equation

The left side will simplify because division and multiplication by 8 will cancel out: $$ x = 96 \times 8 $$

Step 4: Calculate the value of x

Multiplying 96 by 8, we get: $$ x = 768 $$ Now the equation is solved, and the value of x is x = 768.

Key Concepts

Isolate Variable

When you encounter an equation, think of it as a balance scale. On one side, you have the variable you need to find the value for, and on the other, the known quantities. To find the value of the variable, your first task is to 'isolate' it, which means getting the variable on one side of the equation all by itself.

For instance, in the equation \(\frac{x}{8}=96\), \(x\) is trapped within a division by 8. To free \(x\) and isolate it, you must perform the same mathematical operation on both sides of the equation to maintain the balance. Isolating the variable is the crucial first step in solving an equation because it simplifies the problem down to the basic operation needed to find the variable's value.

Inverse Operations

The secret behind isolating the variable is using 'inverse operations'. These are pairs of operations that reverse, or 'undo', each other. Think of them as a lock and key system; one operation locks, and the other unlocks.

Common inverse operations you'll encounter in algebra include addition and subtraction, as well as multiplication and division. For example, if a variable is multiplied by a number, you divide both sides of the equation by that same number to 'unlock' the variable. In the exercise \(\frac{x}{8}=96\), multiplication by 8 is the inverse operation that cancels out division by 8, thus isolating \(x\) on one side of the equation.

Simplify Equation

Simplifying an equation means to make it as easy as possible to understand or solve. This involves combining like terms, cancelling out operations, and sometimes factoring.

Let's use our equation as an example again: when we perform the inverse operation and multiply both sides by 8, we're left with \(8 \times \frac{x}{8} = 768\). Even though it seems we've made the equation more complex, we've actually set up \(x\) to be freed. Since \(8\) divided by \(8\) equals \(1\), they cancel each other out. So the equation simplifies to just \(x = 768\), which gives us the clear solution we were looking for.

Mathematical Calculations

The final and sometimes most satisfying part of solving an equation is performing the mathematical calculations to find the exact numerical value of the variable.

In our example, this step involves the multiplication of 96 by 8 to find the value of \(x\). This straightforward calculation, \(96 \times 8 = 768\), provides us with the precise answer. While the multiplication and division might be straightforward here, calculations can involve a variety of mathematical operations, depending on the complexity of the equation, and may sometimes require a calculator or other computational tools to ensure accuracy.