Problem 12 - College Algebra with Corequisite Support

Question

For each of the following exercises, solve the equation for y in terms of \(x\). $$ 3 x-2 y=6 $$

Step-by-Step Solution

Answer

The equation solved for \(y\) is \(y = \frac{3}{2}x - 3\).

Step 1: Identify equation variables

The given equation is \(3x - 2y = 6\). We need to solve for \(y\) in terms of \(x\), which means expressing \(y\) on one side of the equation.

Step 2: Isolate term with y

To isolate the term with \(y\), subtract \(3x\) from both sides of the equation. This gives us:\[-2y = -3x + 6\]

Step 3: Solve for y

Divide every term in the equation \(-2y = -3x + 6\) by \(-2\) to solve for \(y\):\[y = \frac{-3x + 6}{-2}\]

Step 4: Simplify the expression

Simplify the expression by dividing each term inside the fraction by \(-2\):\[y = \frac{3}{2}x - 3\]

Key Concepts

Solving Equations

When we talk about solving equations, we mean finding the value of variables that make the equation true. An equation is like a balance scale: whatever we do to one side, we must do to the other.
In the exercise, we start with the linear equation: \[3x - 2y = 6\] Our goal is to solve for \(y\) using \(x\).

• First, recognize which variable you need to solve for. In our case, it's \(y\).
• Keep the equation balanced by performing the same operation on both sides.
• Once balanced, you can proceed to the next step, slowly unraveling the equation until you have \(y\) by itself.

Every step you take helps you to understand the relationship between \(x\) and \(y\). It tells us exactly how to change \(x\) to find \(y\).

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations. This step is crucial when working with any kind of mathematical problems. It's about changing the equation without changing its value.
Consider the given equation: \[3x - 2y = 6\] To isolate \(y\), first subtract \(3x\) from both sides to move terms around while maintaining equality: \[-2y = -3x + 6\]

• Understand the need to make strategic moves – like moving \(3x\) to the other side. This makes working on \(y\) exclusively easier.
• Simplify each term as much as possible. It's all about making the equation simpler to handle.
• Remember, the overall objective is to get \(y\) on its own with minimal hassle.

By mastering algebraic manipulation, you can easily transform complex equations into simpler forms without altering their intrinsic meaning.

Isolating Variables

Isolating variables is the key to solving equations effectively. It means getting the variable you’re solving for all alone on one side of the equation. This isolates it and makes it easy to see the relationship it has with other variables.
In our equation, once we have: \[-2y = -3x + 6\] The next goal is to express \(y\) alone, which requires dividing each term by \(-2\):
\[y = \frac{-3x + 6}{-2}\] This results in: \[y = \frac{3}{2}x - 3\]

• Always perform the same math operation on each part of the equation. This maintains the balance.
• Breaking down complex fractions by simplifying each term can reveal easier-to-understand solutions.
• Comprehending the role of each operation helps in better visualization of the equation.

Learning to isolate variables equips students with a powerful tool to solve various mathematical problems with confidence.