Problem 92 - Algebra 1

Question

Solve the equation. \(9 x=3\)

Step-by-Step Solution

Answer

The solution to the equation \(9x = 3\), therefore, is \(x = 0.33\).

Step 1: Analyze the Equation

The equation provided is \(9x = 3\). In this equation, \(x\) is the unknown variable that needs to be solved. Clearly, \(x\) is being multiplied by 9.

Step 2: Isolate the Variable

To isolate \(x\), it is necessary to counteract the multiplication operation that is currently being applied to it. This action can be achieved by dividing both sides of the equation by 9. Now, the equation transforms to \(x = 3/9\).

Step 3: Simplify the Equation

After performing the division operation, the derived fraction can be simplified for easy understanding. The equation becomes \(x = 3 ÷ 9 = 0.333...\) which can be approximated to \(x = 0.33\) to two decimal places for simplicity.

Key Concepts

Algebra Concepts

In algebra, equations are key to finding unknown values. These equations often contain variables, like \(x\), which represent the unknowns. When you encounter an equation, like \(9x = 3\), the objective is to find the value of \(x\). This involves rearranging the equation so that \(x\) is isolated.
This rearrangement process requires understanding how to manipulate mathematical operations to maintain the balance of the equation on both sides. Understanding the role of variables and constants (known numbers like 9 and 3 here) is fundamental in solving algebraic equations.

Equation Simplification

Simplifying an equation means making it as clear and easy to understand as possible. For the equation \(9x = 3\), simplification occurs after isolating \(x\). Once you have isolated \(x\), you simplify further by performing any additional calculations needed, such as division or reducing fractions to their simplest form.
This is done to make it more straightforward for interpretation and understanding. Simplified equations are easier to use for further calculations or for checking solutions efficiently. Maintaining simplicity helps in verifying whether the solution is correct by substituting back the value into the original equation.

Fractions in Equations

Fractions often appear in algebraic equations. When solving \(9x = 3\) and isolating \(x\), you perform the division \(3/9\). This results in the fraction \(\frac{1}{3}\). Learning how to handle fractions is crucial. This includes knowing how to simplify them, such as reducing \(\frac{3}{9}\) to \(\frac{1}{3}\).
Simplifying fractions makes solving and understanding an equation easier. Recognizing equivalent fractions, like knowing \(\frac{1}{3}\) is the same as repeating decimal \(0.333...\), is another valuable skill in algebra. By simplifying, you lessen the complexity and make the answers cleaner and more digestible.