Problem 1 - Elementary Algebra

Question

Solve \(6 a=42\) for \(a\).

Step-by-Step Solution

Answer

Answer: \(a = 7\)

Step 1: Divide both sides by 6

First, we need to isolate the variable \(a\) by dividing both sides of the equation by 6. This will give us: \[a = \frac{42}{6}\]

Step 2: Simplify the expression

Now we need to simplify the expression on the right-hand side by dividing 42 by 6: \[a = 7\] So, the solution to the equation \(6 a = 42\) is \(a = 7\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is key to unlocking the mysteries of algebra. An algebraic expression is a combination of numbers, variables (such as a), and operation symbols like addition (+), subtraction (-), multiplication (*), and division (/). In our exercise, 6a=42, 6a is an algebraic expression where 6 is a coefficient that multiplies the variable a.

These expressions can become more complex with the addition of more variables and numbers. When faced with such expressions, it's crucial to understand how to manipulate them to solve for a variable. Recognizing that each part of the expression plays a role in reaching the solution fosters a deeper comprehension of the problem-solving process.

Isolating Variables

When it comes to solving linear equations, one of the fundamental skills is isolating the variable. This means manipulating the equation in such a way that you get the variable by itself on one side of the equation. It's like finding the star of the show and giving it its own spotlight.

For instance, in our exercise, isolating the variable a involves getting rid of the coefficient 6 attached to it. This is typically done through inverse operations: if the variable is multiplied by 6, we divide by 6 on both sides to cancel it out. It's a bit like having a balance scale where you must do the same thing to both sides to keep it level. Once isolated, a stands alone, and the equation becomes straightforward to solve.

Simplifying Expressions

After isolating variables, the next step is simplifying the expression to find the simplest form of the solution. Simplifying an expression means reducing it to its most basic terms. In arithmetic terms, this may include basic operations like adding, subtracting, multiplying, and dividing.

In our textbook exercise of 6a=42, once we've isolated a to one side, we're left with an expression that can be simplified: a = 42/6. By performing the division, we simplify the expression to the most reduced form, which in this case is a = 7. Simplifying expressions helps in clearly understanding the value of the variables and it's essential for checking the solution's accuracy.