Problem 22 - Elementary Algebra

Question

\(5 x+9 y=17\) for \(y\)

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = \frac{17 - 5x}{9}\).

Step 1: Identify the equation

We are given the equation: \(5x + 9y = 17\). Our goal is to solve this equation for \(y\).

Step 2: Isolate the term with y

To isolate the term with \(y\), we need to get \(9y\) by itself on one side. Start by subtracting \(5x\) from both sides of the equation: \[9y = 17 - 5x\].

Step 3: Solve for y

Now, we divide every term in the equation by 9 in order to solve for \(y\). This gives us: \[y = \frac{17 - 5x}{9}\].

Key Concepts

Solving for a Variable

In mathematics, solving for a variable involves finding the value of a specific variable within an equation. The objective is to rewrite the equation so that the variable appears on one side of the equation by itself. This process helps us determine the specific value or expression for the variable we are interested in.
In the exercise provided, the goal is to solve the equation \(5x + 9y = 17\) for \(y\). To "solve for \(y\)", we aim to express \(y\) in terms of \(x\) and any constants. This means that everything other than \(y\) will be moved to the other side of the equation, ensuring that \(y\) is isolated and can be examined independently. This is central to understanding relationships between different variables and is a crucial skill in algebra.

• Rewriting equations to find one specific variable.
• Understanding the relationship between variables.
• Isolating the variable of interest for clarity.

Once solved, it allows for deeper insights into the equation's behavior and relationships.

Isolation of Terms

Isolation of terms is an essential step when solving equations for a specific variable. This involves manipulating the equation so that the term containing the variable is isolated on one side. This step is crucial because it enables us to see how the variable interacts with the rest of the equation.
In the original exercise, we need to isolate the term with \(y\) in the equation \(5x + 9y = 17\). To do this, we subtract \(5x\) from both sides, resulting in \(9y = 17 - 5x\). This step is vital because it removes any additional components attached to \(y\), making it easier to perform further operations that will solve for \(y\).
Here are some key points:

• Identify terms that include the variable of interest.
• Use inverse operations (e.g., subtraction for addition) to cancel out other terms.
• Maintain the equation's balance by performing the same operation on both sides.

These steps ensure that we systematically approach the goal of isolating the desired variable, setting the stage for further simplification.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations using a variety of algebraic operations. This skill is fundamental in solving equations, as it allows us to express variables in different configurations.
In our exercise, after isolating \(9y\) by subtracting \(5x\), we then perform division to solve for \(y\). Each side of the equation \(9y = 17 - 5x\) is divided by 9, resulting in \(y = \frac{17 - 5x}{9}\). This step demonstrates algebraic manipulation where we systematically use division to simplify the equation and express \(y\).
Some key algebraic manipulation skills include:

• Applying inverse operations to isolate the variable.
• Ensuring operations maintain the equality of the equation.
• Simplifying expressions to make them more comprehensible.

Mastering these operations not only helps solve equations but is beneficial in understanding complex mathematical concepts.