Problem 26 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Use division to solve the equation. $$ 8 x=3 $$

Step-by-Step Solution

Answer

The solution to the equation \(8x = 3\) is \(x = \frac{3}{8}\).

Step 1: Understand the equation

The equation given is \(8x = 3\). Here, \(x\) is multiplied by \(8\) and the result is \(3\). The objective is to find the value of \(x\).

Step 2: Isolate the variable x

To isolate \(x\), divide both sides of the equation by \(8\). Doing this ensures the equation's balance is maintained. This results in \(x = \frac{3}{8}\).

Key Concepts

Division in Equations

When solving linear equations, one of the core techniques used is division. Division helps us manipulate an equation to isolate a variable, making it easier to find its value.
In the equation \(8x = 3\), \(x\) is multiplied by 8. To find what \(x\) equals, we need to do the opposite of multiplying by 8.

The opposite action is dividing by 8. By dividing both sides of the equation by 8, each term is still equal, keeping the equation balanced:

• On the left side, \(8x\) divided by 8 simplifies to \(x\).
• On the right side, \(3\) divided by 8 stays as \(\frac{3}{8}\).

Using division in this way allows us to isolate the variable and find its value.

Isolating Variables

Isolating the variable is a crucial step in solving equations. We aim to get the variable by itself on one side of the equation.
This shows us exactly what the variable equals.

In the equation \(8x = 3\), our goal is to get \(x\) by itself. Since \(x\) is being multiplied by 8, we use division—its opposite operation—to isolate \(x\).
By dividing both sides of the equation by 8, we effectively cancel the multiplication, leaving:

• \(x = \frac{3}{8}\)

Once isolated, the variable tells us its value or represents its relationship to the other terms in the equation.
This ability to manipulate and isolate variables is fundamental to solving any equation.

Fractional Solutions

In many cases, solving equations leads to solutions that are fractions.
This means that instead of a simple whole number, the solution is expressed as the ratio of two numbers.

Let's consider the equation \(8x = 3\).
After dividing both sides by 8, we find \(x = \frac{3}{8}\).
This fraction \(\frac{3}{8}\) is the exact representation of \(x\)'s value.

Working with fractions might seem tricky at first, but it often simplifies the understanding and manipulation of numerical relationships:

• Fractions represent parts of a whole.
• They ensure precision and exactness in solutions.

Learning to work comfortably with fractional solutions is an essential skill in mathematics.
It ensures that you can handle various types of equations smoothly and efficiently.