Problem 66 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ a x+b y=c \quad \text { for } y $$

Step-by-Step Solution

Answer

The solution for \( y \) is \( y = \frac{c - ax}{b} \).

Step 1: Identify the Goal

We want to solve the given equation for variable \( y \). This means we need to manipulate the equation so that \( y \) is isolated on one side.

Step 2: Move the Term with y

The given equation is \( ax + by = c \). Our goal is to isolate \( y \). Start by subtracting \( ax \) from both sides to move terms not involving \( y \). This gives us: \[ by = c - ax \]

Step 3: Isolate y

Now, to solve for \( y \), divide every term by \( b \), as \( b \) is the coefficient of \( y \). This isolates \( y \) and presents the solution: \[ y = \frac{c - ax}{b} \]

Step 4: Verify the Solution

Ensure that the solution makes sense by comparing it to the original equation. Since we performed algebraic operations correctly (subtraction and division), the solution \( y = \frac{c - ax}{b} \) is consistent. It expresses \( y \) purely in terms of \( a, x, c, \text{ and } b \).

Key Concepts

Isolation of Variables

The isolation of variables is a fundamental technique used in solving equations, particularly in algebra. When we isolate a variable, we rearrange the equation so that the desired variable is by itself on one side of the equation. This is essential for finding the value or expression for that particular variable.

In our exercise, the variable we want to isolate is \( y \). The given equation is \( ax + by = c \). Our goal is to rearrange it to get \( y \) by itself. We achieve this by eliminating or "moving" other terms away from \( y \), using inverse operations.

• Step 1: Identify what we need to solve for. Here, it's \( y \).
• Step 2: Move all other terms away. We subtract \( ax \) from both sides.
• Step 3: Solve for \( y \) by dividing by its coefficient.

Isolation is crucial because it simplifies complex equations into understandable parts, making it easier to solve and interpret.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to transform an equation. These manipulations involve addition, subtraction, multiplication, division, and other operations to both simplify and solve equations. Each step logically follows from the previous, ensuring equations remain balanced and true.

In the equation \( ax + by = c \), we perform a sequence of manipulations. First, to isolate \( y \), we subtract \( ax \) from both sides. This creates an equation where \( by = c - ax \). The next manipulation is dividing everything by \( b \) to solve for \( y \), giving us \( y = \frac{c - ax}{b} \).

Each step follows these guiding principles:

• Operations performed on one side must be done to the other.
• Inverse operations are used to cancel terms (e.g., subtract to cancel an addition).
• Maintaining the equation's balance is crucial for the solution to remain valid.

Algebraic manipulation is the key to solving and simplifying complex equations.

Equations in Two Variables

Equations in two variables are mathematical expressions that require solutions for both variables involved. These are often written in the standard form \( ax + by = c \). Many real-life problems can be modeled as such equations, making understanding them vital.

In these equations, typically, we solve for one variable in terms of the other. This ability allows us to substitute values and understand the relationship between variables. Using our example \( ax + by = c \), isolating \( y \) provides insight into how \( y \) changes with respect to \( x \), \( a \), \( b \), and \( c \).

Key reasons to learn about these types of equations:

• They provide a method to understand relationships between different quantities.
• They are a gateway to understanding more complex systems in algebra and calculus.
• Solving them aids in modeling and predicting outcomes in numerous fields like physics, economics, and engineering.

Mastering equations in two variables builds a strong foundation for exploring advanced mathematical concepts.