Problem 29 - Algebra 1: Concepts and Skills

Question

Rewrite the equation in function form. $$ 3 x+2 y=-3 $$

Step-by-Step Solution

Answer

The function form of the given equation is \(y = -3/2 - 3x/2\).

Step 1: Original Equation

The given equation is \(3x + 2y = -3\).

Step 2: Isolate the y term

To write this in function form, one must isolate \(y\). Start by moving the term \(3x\) from the left-hand side to the right-hand side of the equation to get: \(2y = -3 - 3x\).

Step 3: Solving for y

Next, one needs to divide each term on both sides of the equation by \(2\) to solve for \(y\), resulting in: \(y = -3/2 - 3x/2\).

Key Concepts

Rewriting Equations

Rewriting equations is all about transforming them into a different, often more useful, format. Consider the equation you start with: \[ 3x + 2y = -3 \] This includes both an \(x\) term and a \(y\) term on the same side of the equation. The goal here is to rearrange everything to bring clarity or to express one variable distinctly in terms of others.

• Begin by recognizing the terms and constants involved.
• Determine which variable you need to isolate (in function form, it's often the dependent variable, like \(y\)).

In this manner, you simplify and reorganize the equation, making it easier to work with in various contexts like graphing or further solving.

Isolating Variables

Isolating a variable involves getting one particular variable alone on one side of the equation, leaving everything else on the opposite side. In our equation,\[ 2y = -3 - 3x \]you want to isolate \(y\). This step is crucial for expressing one variable clearly in terms of others. Here's how you can achieve this:

• Identify which variable you want to isolate - in this case, \(y\).
• Move other terms to the opposite side by performing operations like addition, subtraction, multiplication, or division.

Start by removing the \(3x\) term from the left side by subtracting it from both sides. You achieve:\[ 2y = -3 - 3x \]By staying methodical, isolating becomes straightforward, paving the way to solve specifically for whichever variable you're targeting.

Solving for y

To solve for \(y\) means to manipulate the equation until \(y\) stands by itself. Once you've isolated the \(y\) term, you're halfway there. Taking the isolated equation:\[ 2y = -3 - 3x \]the next move is clear: divide each term by the coefficient of \(y\), which in this instance is 2. This step ensures \(y\) is fully isolated:\[ y = \frac{-3}{2} - \frac{3x}{2} \]

• Divide every term by the coefficient of \(y\) to ensure clarity.
• This method transforms the equation into a direct expression for \(y\), often useful for graphing equations in the form \(y=mx+b\).

Completing this process not only gives you a clear expression of \(y\) but also prepares the equation for various applications, such as plotting its graph on a coordinate plane.