Problem 26 - Introductory Algebra for College Students

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A x+B y=C\) for \(y\)

Step-by-Step Solution

Answer

The formula solved for \(y\) is \(y = \frac{C - Ax}{B}\)

Step 1: Isolate the y-variable

Begin by isolating y on one side of the equation. To do this, subtract \(Ax\) from both sides of the equation to get rid of it from left hand side. The equation now becomes: \(B y = C - Ax\)

Step 2: Solve for y

Next, divide both sides of the equation by \(B\) to solve for \(y\). Doing so yields the final equation: \(y = \frac{C - Ax}{B}\)

Key Concepts

Solving Equations

In algebra, one of the most crucial skills a student can acquire is the ability to solve equations. A linear equation like \(A x + B y = C\) represents a relationship between two variables. Solving these equations involves finding the values of the variables that make the equation true.

The general approach to solving equations is to manipulate them in such a way that you isolate the desired variable on one side of the equation.Students solve equations by performing the same mathematical operations on both sides of the equation, maintaining the balance, or equality, between them.

• Start by simplifying both sides of the equation if necessary.
• Use addition or subtraction to eliminate terms on one side.
• Use multiplication or division to isolate the variable.

Understanding this process is foundational for success in algebra and beyond.

Isolating Variables

Isolating variables is a key step when solving any equation. This means getting the variable alone on one side of the equation. For example, if given the equation \(A x + B y = C\) and asked to solve for \(y\), we need to isolate \(y\) first.

Here’s how you can isolate a variable step by step:

• Identify the term with the variable you want to solve for (in this case, \(B y\)).
• Move all other terms to the opposite side by adding or subtracting. In the example, subtracting \(A x\) from both sides gives \(B y = C - A x\).
• Finally, to completely isolate \(y\), divide each term by \(B\), resulting in \(y = \frac{C - A x}{B}\).

By following these steps, isolating the variable becomes a straightforward task.

Formulas in Algebra

Formulas in algebra are equations that express a relationship between different quantities. They are like recipes that instruct us on how to calculate or find a specific value given certain variables.

A linear equation such as \(A x + B y = C\) can be seen as a formula where \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables.

• This type of equation might represent a line on a graph.
• By solving the formula for one of the variables, you can find how changes in another affect it.

Knowing how to rearrange formulas to solve for any variable is essential for tackling more complex problems in algebra and other mathematics fields.
Learning to recognize and manipulate these relationships helps to deepen understanding and open up solutions to a broad range of problems.