Problem 73 - Algebra 1: Concepts and Skills

Question

solve the equation. (Lessons 3.1, 3.2, and 3.3) $$\frac{a}{4}=3$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{a}{4}=3\) is \(a=12\).

Step 1: Identify the Problem

The task is to solve the equation \(\frac{a}{4}=3\) for \(a\). This is a simple linear equation where \(a\) is divided by 4.

Step 2: Multiply Both Sides by 4

To solve for \(a\), multiply both sides of the equation by 4. This is done to cancel out the denominator on the left side. Hence, on multiplying, we get \(4*\frac{a}{4}=4*3\).

Step 3: Simplify the Equation

After carrying out the multiplication, the equation simplifies to \(a=12\).

Key Concepts

Solving Equations

Linear equations are a fundamental part of algebra. When you solve them, you're finding what value makes an equation true. It's like your secret detective mission to find the missing piece! The main objective is to isolate the variable, often represented by a letter such as "a" in our case, on one side of the equation.
When faced with an equation like \( \frac{a}{4} = 3 \), your goal is to figure out what value "a" must have to keep both sides balanced. Balancing is a crucial concept; it ensures the equation's equality is maintained while you work toward finding that secret number for "a".

• Identify the variable: Determine which letter or symbol represents the unknown number you need to solve for. In our exercise, it's "a".
• Equivalence principle: Keep the equation balanced. Whatever you do to one side, you must do to the other.

Ensuring each side remains equal as you solve takes some practice but becomes easier with these guiding principles.

Algebraic Expressions

Algebraic expressions are combinations of letters, numbers, and operations that stand for a certain quantity. They are essential in forming and solving equations. In our example equation, \( \frac{a}{4} \), you see a fraction, where "a" is divided by 4.
Your mission is to understand this expression in pieces. It's telling us that some mystery number "a" has been split into 4 equal parts, and this entire expression equals 3.

• Understanding terms: Terms are parts of an expression separated by addition or subtraction signs. Here "\( \frac{a}{4} \)" is a single term where division is the key operation.
• Recognizing operations: Knowing operations like addition, subtraction, multiplication, and division helps in breaking down expressions. "\( \frac{a}{4} \)" involves division.

Identifying how expressions relate to the scenario increases your confidence in manipulating and simplifying them to solve equations.

Multiplication in Equations

Multiplication is a powerful tool in solving equations, especially when dealing with fractions. It can help eliminate division, making the path to the solution straightforward. In the equation \( \frac{a}{4} = 3 \), we use multiplication to rid the fraction of its divisor.
To solve for "a", multiply both sides of the equation by 4, like this:
\[ 4 * \frac{a}{4} = 4 * 3 \]
This step effectively cancels out the 4 in the denominator, simplifying to just "a". On the right, multiplying gives us 12. Thus, ensuring that both sides are still in balance while removing the division.

• Canceling out variables: Multiplication can help eliminate divisors, simplifying the equation.
• Maintaining balance: It's crucial to multiply each side by the same number to keep the equation balanced.

By being mindful of these principles, multiplication serves as an ally in solving complex-looking equations swiftly.