Problem 58 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ 3 x-5 y=19 \quad \text { for } y $$

Step-by-Step Solution

Answer

The solution is \(y = \frac{3}{5}x - \frac{19}{5}\).

Step 1: Move Terms Involving y

We start with the given equation: \(3x - 5y = 19\). The goal is to solve for \(y\), so we need to isolate \(y\) on one side of the equation. This can be done by first subtracting \(3x\) from both sides of the equation. \[-5y = -3x + 19\]

Step 2: Solve for y

Next, divide each term in the equation \(-5y = -3x + 19\) by \(-5\) to isolate \(y\). \[y = \frac{-3x}{-5} + \frac{19}{-5}\] Simplifying the expression gives \[y = \frac{3}{5}x - \frac{19}{5}\]

Key Concepts

Isolating Variables

When solving an equation for a specific variable, your main task is to "isolate" that variable on one side of the equation. This means making the target variable the only term on one side, while all other terms are moved to the opposite side.
To begin isolating the variable, identify the variable you want to solve for. For instance, in the equation \(3x - 5y = 19\), you want to solve for \(y\). Recognize that you need all terms involving \(y\) on one side.

• Move any terms not involving \(y\) to the other side by using addition, subtraction, multiplication, or division.
• Apply these operations to each side of the equation to maintain equality.

In our specific example, subtracting \(3x\) from both sides helps start this process: \[-5y = -3x + 19\]. It ensures each maintain balance while edging closer to isolating \(y\).

Equation Simplification

Simplification is reducing an equation into its simplest form, making it easier to analyze or solve. Once you've isolated the term with the variable you're solving for, simplification helps clarify each part of the equation.
In the simplified version of our example, \[-5y = -3x + 19\], simplification occurs when balancing - or rearranging - to make working with the equation easier.

• Identify common factors or combine like terms, if possible.
• Use division or multiplication to eliminate numbers next to variables, which simplifies terms fully.

To solve for \(y\), dividing everything by \(-5\) simplifies the equation to \[y = \frac{3}{5}x - \frac{19}{5}\]. This transforms the equation into a clear, straightforward line in slope-intercept form.

Algebraic Manipulation

Algebraic manipulation involves operations and properties of equalities to rearrange equations in the desired form. It's essential for translating studies into real-world applications.
This technique can include combining like terms or using properties like the distributive property to expand or simplify expressions. Each algebraic manipulation step should keep both sides of the equation balanced and equivalent to the original.

• Utilize addition and subtraction to move terms between sides of the equation.
• Apply multiplication or division to isolate and solve for variables without altering the equation's balance.

For our exercise, we've applied multiplication and division effectively to isolate \(y\). After subtraction, dividing each term by \(-5\) is a form of algebraic manipulation that isolates \(y\), giving us a manageable form:\[y = \frac{3}{5}x - \frac{19}{5}\]. This not only solves for \(y\) but reflects the shift from a complex system to straightforward results.