Problem 26 - Algebra for College Students

Question

Solve each equation for \(y\). $$3 x+y=9$$

Step-by-Step Solution

Answer

y = 9 - 3x

Step 1: Rewrite the given equation

Start by writing down the given equation: \[ 3x + y = 9 \]

Step 2: Isolate the variable y

To solve for \( y \), we need to isolate it on one side of the equation. Subtract \( 3x \) from both sides of the equation to get: \[ y = 9 - 3x \]

Key Concepts

Isolating Variables

When solving linear equations, the first key step is often isolating the variable you are solving for. In our example, we need to isolate the variable \(y\) in the equation \(3x + y = 9\). This means getting \(y\) by itself on one side of the equation. Here's how you can do that:
Start with the equation \[3x + y = 9\]
To isolate \(y\), you need to remove \(3x\) from the left side of the equation. You can do this by subtracting \(3x\) from both sides:

• \(3x + y - 3x = 9 - 3x\)

This simplifies to:

• \( y = 9 - 3x \)

Now, \(y\) is isolated, which means it is alone on one side of the equation. Isolating variables is a core concept in algebra because it allows you to directly solve for the unknown.

Solving for y

Solving for \(y\) in a linear equation means manipulating the equation so that \(y\) ends up by itself on one side. Let's work through this with our equation: \(3x + y = 9\). We aim to isolate \(y\), and we have already seen how to do this by subtracting \(3x\) from both sides:

• \( y = 9 - 3x \)

This tells us that \(y\) equals \(9\) minus three times \(x\).
This representation of \(y\) in terms of \(x\) is very useful. It shows the direct relationship between the two variables. Understanding how to solve for a specific variable allows you to analyze and understand how changes in one variable affect the other.

Basic Algebra

Basic algebra involves foundational techniques and operations that are essential for solving equations. One of the principles we use here is the balance method: whatever you do to one side of the equation, you must do to the other. In our equation \(3x + y = 9\), to isolate \(y\):

• First, recognize that \(3x\) is added to \(y\).

To undo this addition, you do the opposite operation, which is subtraction. Subtract \(3x\) from both sides:

• \( y = 9 - 3x \)

This process of isolating a variable by performing the same operation on both sides is a basic algebraic technique used to maintain the balance of the equation while transforming it into a more solvable form. Remember, algebra is like a puzzle, where each step brings you closer to finding the value of the unknown variables. By practicing these steps, you'll gain confidence in solving more complex equations in the future.