Problem 5 - Algebra 1: Concepts and Skills

Question

Solve the equation for the indicated variable. $$3 y=x ; y$$

Step-by-Step Solution

Answer

The solution for y is \(y = \frac{x}{3} \).

Step 1: Given Equation

The given equation is \(3y = x\). We need to solve it for 'y'.

Step 2: Isolate the Variable 'y'

To isolate 'y', we need to divide both sides of the equation by 3. This gives us \[y = \frac{x}{3} \].

Key Concepts

Isolating the Variable

In the world of algebra, one crucial skill is isolating the variable you're solving for. This basically means getting the variable all by itself on one side of the equation.

For example, consider the equation given: \(3y = x\). Your task is to solve for 'y'. So, the goal is to have 'y' appear alone on one side of the equation. To achieve this, you'll need to perform operations that simplify the equation, ultimately leaving 'y' by itself.

Remember: isolating a variable often involves performing the same operation on both sides of an equation. This keeps the equation balanced, just like a perfectly even scale! Keep this in mind as we dive deeper into the mechanics of algebraic manipulation.

Algebraic Manipulation

Once you aim to isolate a variable, algebraic manipulation is your go-to tool. This involves using operations like addition, subtraction, multiplication, and division to rearrange equations.

In the equation \(3y = x\), our main focus is freeing 'y' from multiplication by 3. To manipulate this algebraically, you divide every term by 3:

\[ \frac{3y}{3} = \frac{x}{3} \]

Here, division by 3 effectively cancels the coefficient of 'y' on the left side, leaving \(y\) by itself. These operations allow for simplifying equations while maintaining their equality.

Tips for algebraic manipulation include:

• Always perform the same operation on both sides of the equation.
• Be mindful of operations that undo each other, such as multiplying and dividing.
• Check your work by plugging values back in to ensure consistency.

Solving Equations

Finally, solving equations brings everything together. You've isolated the variable, manipulated the equation properly, and now it's about finding the solution. The solution is the value that makes the equation true.

In our simplified form \(y = \frac{x}{3}\), 'y' represents any number that keeps the equation in balance when you know the value of 'x'. This step is where you'd substitute known values to evaluate or further analyze the solution.

Equation solving is critical because it uncovers relationships and answers that are hidden within mathematical statements. Here are some key points on solving equations:

• Ensure that your equation is correctly simplified before solving.
• Recheck algebraic steps to avoid simple mistakes.
• Use the solution to check your work by substituting back into the original equation.

Understanding these essential concepts empowers you to tackle more complex problems with confidence.