Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation for the indicated variable. $$ a x+b y+c=0 ; y $$

Step-by-Step Solution

Answer

The solution to the given equation \(ax + by + c = 0\) for the variable \(y\) is \(y = \frac{-ax - c}{b}\).

Step 1: Arrange the equation to isolate y

First, rearrange the equation in order to isolate the term containing \(y\) on one side. To do this, subtract \(ax\) and \(c\) from both sides of the equation: \[ by = -ax - c \]

Step 2: Solve for y

To isolate \(y\), divide both sides of the equation by \(b\): \[ y = \frac{-ax - c}{b} \] Now that \(y\) is isolated, the equation has been solved for \(y\). So, the final solution is: \[ y = \frac{-ax - c}{b} \]

Key Concepts

Solving Equations

Solving equations is like finding the unknown treasure in a mystery story. The goal is to find the value of a variable that makes the equation true. When you solve an equation, you are balancing both sides to maintain equality. If you imagine a scale, doing something to one side means you must do the same to the other to keep it balanced.
To begin, you should identify the goal. In the given problem, the target is to solve for the variable \( y \). You need to use mathematical operations such as addition, subtraction, multiplication, or division to isolate and solve the variable of interest.
Remember:

• Keep equations balanced by performing operations on both sides.
• Rearrange the equation logically, step by step.

Once the variable is isolated, checking your work by plugging the solution back into the original equation can confirm accuracy.

Linear Equations

Linear equations are the simplest form of equations and can be thought of as straight lines when plotted on a graph. They are called 'linear' because they involve only first-degree terms, which means variables are not raised to any power other than one. This makes them easy to solve, predict, and understand.
In "\( ax + by + c = 0 \)", each term is either a constant or the product of a constant and a single variable:

• \( ax \) and \( by \) are linear terms.
• \( c \) is a constant term without a variable.

Linear equations can represent a straight line in a coordinate plane. The form \( y = mx + b \) is essential, where \( m \) is the slope and \( b \) is the y-intercept. Understanding how to rearrange an equation into this form is crucial for graphing or finding solutions.

Variable Isolation

Variable isolation is a fundamental concept in algebra. It refers to the process of manipulating an equation so that the variable you are solving for is by itself on one side of the equation. Think of it as putting the spotlight on the variable you need.
In our exercise, we isolated \( y \). We rearranged the equation to place all other terms on the opposite side:

• Subtract terms not involving \( y \) first, like \( ax \) and \( c \).
• Divide by the coefficient of \( y \), which is \( b \), to have \( y \) by itself.

If you follow these steps, you can solve any equation for the given variable. This technique is widely used and forms the backbone of algebra.
Mastering the art of variable isolation will aid in effectively tackling more complex algebraic equations, strengthening your mathematical skill set.