Question

Solve the equation. $$ \frac{a}{4}=3 $$

Short answer

The solution to the equation is \(a = 12\).

Step-by-step solution

Isolate the variable

The aim is to isolate \(a\) on one side of the equation. Currently, \(a\) is divided by 4. To undo this division, multiply both sides of the equation by 4. After this operation, the equation becomes: \(a = 4 \cdot 3\).

Simplify the right side of the equation

Calculate the value on the right side of the equation: \(4 \cdot 3 = 12\). So, \(a = 12\).

Key concepts

Isolate the Variable

Understanding how to isolate the variable in an equation is a cornerstone of algebra. In our example, the goal is to find the value of the variable 'a' by getting it alone on one side of the equation. To do this, you usually need to perform the inverse (opposite) operation of what is currently being applied to the variable. Here, the variable 'a' is divided by 4, so we multiply each side of the equation by 4 to cancel out the division. This leaves 'a' by itself, successfully isolating it. Remember, whatever you do to one side of the equation, you must do to the other to keep it balanced.

Simple Algebraic Operations

The term simple algebraic operations refers to basic math processes like addition, subtraction, multiplication, and division. In our exercise, once we've isolated the variable by multiplying both sides by 4, we then need to simplify the right side of the equation using multiplication. These operations are the building blocks of algebra and are essential for solving equations. When multiplying or dividing, remember the properties of operations—multiplication and division are reciprocal processes. By mastering these simple operations, you'll be able to manipulate and solve a wide variety of algebraic expressions.

Equation Simplification

Equation simplification is about making an equation easier to understand and solve. Simplifying an equation can involve combining like terms, reducing fractions, or performing simple algebraic operations to remove complex expressions. In this case, once we multiply the left side by 4, the 'a' stands alone without any coefficients or constants attached to it. Then, by calculating the product of 4 and 3 on the right side, we simplify the equation to its most basic form, which gives us the solution 'a = 12'. Remember, the goal of simplification is clarity and ease in finding the solution of the variable.