Problem 40 - Algebra for College Students

Question

$$ 3 x-2 y=-1 \quad \text { for } y $$

Step-by-Step Solution

Answer

The solution for \(y\) is \( y = \frac{3}{2}x + \frac{1}{2} \).

Step 1: Isolate the y-term

Start with the given equation: \(3x - 2y = -1\). In order to solve for \(y\), we need to isolate the \(y\)-term. Step one is to move the \(3x\) to the other side of the equation by subtracting \(3x\) from both sides: \(-2y = -1 - 3x\).

Step 2: Solve for y

Now that the \(y\)-term is isolated on one side, solve for \(y\) by dividing every term by \(-2\) to get \(y\) by itself. Thus, \( y = \frac{-1 - 3x}{-2}\).

Step 3: Simplify the Equation

Simplify the equation by dividing each term in the numerator by \(-2\), which gives us: \( y = \frac{1}{2} + \frac{3}{2}x\). This is the final simplified expression for \(y\).

Key Concepts

Understanding Linear Equations

Imagine you're dealing with equations that involve variables, like little puzzles waiting to be solved. Linear equations are equations where the highest power of the variable is one. These are the simplest kind of equations and form the foundation for more advanced algebra. In our example, the equation is:- \[3x - 2y = -1\]This equation is called "linear" because both \(x\) and \(y\) have an exponent of 1. Linear equations like this one are straight lines when graphed on a coordinate plane. This kind of problem involves finding the value of one variable when another is already provided or through rearranging the equation.

Isolation of Terms

To solve an equation, we need to "isolate" the term we're interested in. In our exercise, that term is \(y\). Starting with the equation \[3x - 2y = -1\], we need \(y\) by itself on one side of the equation:- First, we subtract \(3x\) from both sides, a key move in the isolation process: \[-2y = -1 - 3x\].- Imagine this as solving a mystery; each term we move or adjust brings us closer to revealing our answer.By isolating \(y\), we simplify the problem—our goal is to express \(y\) in terms of other known quantities. Once \(y\) stands alone, we can clearly see its relationship with \(x\). It's often about keeping balance; whatever you do on one side of an equation, you must do on the other. This keeps the equation valid and allows us to discover new perspectives on old problems.

Simplifying Algebraic Expressions

Now that we have isolated \(y\), we end up with a fraction that looks complicated. This is where simplification helps:- The expression we have is \[y = \frac{-1 - 3x}{-2}\].- Dividing each term by \(-2\) simplifies it: \[y = \frac{1}{2} + \frac{3}{2}x\].Simplifying makes equations easier to interpret. This final form tells us exactly how \(y\) depends on \(x\). Simplification involves canceling out terms, combining like terms, or breaking down fractions. It tidies up an equation and makes it readable and usable in real-world contexts. Always consider simplification as a valuable skill; it clarifies your thinking and streamlines your calculations.