Problem 17 - College Algebra with Modeling and Visualization

Question

Solve the equation for \(x\) and then solve it for \(y .\) $$ \frac{2 x-y}{3 y}=1 $$

Step-by-Step Solution

Answer

The solution is \(x = 2y\); \(y\) can be any non-zero value.

Step 1: Clear the Fraction

To get rid of the fraction, multiply both sides of the equation by the denominator (\(3y\)) to obtain:\[2x - y = 3y\]

Step 2: Rearrange for x

To solve for \(x\), first isolate the term with \(x\) on one side. Start by adding \(y\) to both sides of the equation:\[2x = 3y + y\]Simplify the equation:\[2x = 4y\]

Step 3: Solve for x

Divide both sides of the equation by 2 to solve for \(x\):\[x = \frac{4y}{2}\]Simplify the expression:\[x = 2y\]

Step 4: Substitute x Back to Solve for y

Using the expression \(x = 2y\), substitute \(2y\) into the original equation for \(x\):\[\frac{2(2y) - y}{3y} = 1 \]Simplify the expression inside the numerator:\[\frac{4y-y}{3y} = 1\]\[\frac{3y}{3y} = 1\]

Step 5: Validate and Solve for y

Since the simplified equation is a true statement \(1 = 1\), this confirms our substitutions and simplifications were correct. However, it implies that any value of \(y\) satisfying the original equation satisfies this form. Thus, we can't determine a specific \(y\) unless more information is provided. The relation holds as long as \(x = 2y\).

Key Concepts

Isolating Variables

Isolating variables is a crucial first step when solving algebraic equations. It involves rearranging the equation so that the variable you are solving for is by itself on one side of the equation. This makes it easier to find its value.
In the example provided, we were tasked to find both \(x\) and \(y\). The first step to isolate \(x\) involved manipulating the original equation to remove distractions from \(x\) on its respective side. Here’s how you can do it:

• Identify the term with \(x\). In the equation \(2x - y = 3y\), \(2x\) is the term that contains \(x\).
• Move all other terms to the opposite side of the equation by performing inverse operations. We added \(y\) to both sides: \(2x = 4y\)

This process sets the base to solving any equation, making it easier to target and solve for your goal, the unknown variable.

Substitution Method

The substitution method is often used to solve equations that have multiple variables, by expressing one variable in terms of another. It makes understanding variable relationships easier.
In our solution, once we express \(x\) in terms of \(y\), \(x = 2y\), we can substitute this expression back into the original equation. Here's a walkthrough of how it's done:

• Substitute the known expression \(2y\) for \(x\) in the original equation: \(\frac{2(2y) - y}{3y} = 1\) becomes straightforward to solve.
• By substituting \(x = 2y\), we directly verify the relationship between variables, reducing errors.
• This method can apply when equations give enough information or constraints. Here, it helps to show the dependent nature of each variable.

Substitution helps in cross-verifying solutions and understanding how one variable affects another.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. Using algebraic techniques ensures accuracy and simplifies complex operations.
The solution includes several instances of algebraic manipulation, such as simplifying fractions and combining like terms. Below are some examples:

• Removing the denominator is essential in \(\frac{2x-y}{3y} = 1\), allowing you to multiply through by \(3y\) and clear the fraction entirely. This results in \(2x - y = 3y\).
• Simplifying \(2x = 4y\) by dividing both sides by 2, giving \(x = 2y\), refines the equation to its most simple form.

Algebraic manipulation involves using properties such as distributing, combining like terms, and factoring to tackle any equation effectively. It enhances transparency and understanding of the solution.