Problem 16 - College Algebra

Question

For each of the following exercises, solve the equation for \(y\) in terms of \(x\) . $$5 x+2 y=0$$

Step-by-Step Solution

Answer

The solution is \( y = -\frac{5}{2}x \).

Step 1: Understand the Equation

The given equation is linear: \(5x + 2y = 0\). Our goal is to solve this equation for \(y\) in terms of \(x\), meaning we want \(y\) alone on one side of the equation.

Step 2: Isolate the Term with y

First, we need to move the term involving \(x\) to the other side of the equation. We can do this by subtracting \(5x\) from both sides: \(5x + 2y - 5x = 0 - 5x\). This simplifies to \(2y = -5x\).

Step 3: Solve for y

Now that we have isolated \(2y\), we can solve for \(y\) by dividing each term by 2: \(\frac{2y}{2} = \frac{-5x}{2}\). This simplifies to \(y = -\frac{5}{2}x\).

Key Concepts

Equation Solving

Solving an equation means finding the value(s) of a variable that make the equation true. When working with linear equations, which are equations that graph as a straight line, you aim to express one variable in terms of another. In our original problem, we're given the linear equation \[ 5x + 2y = 0 \] and are tasked with expressing the variable \( y \) in terms of \( x \). Recognizing the type of equation helps in deciding the approach for solving it.

• Linear equations typically have variables added, subtracted, or multiplied by constants.
• They don't involve powers greater than one, products of variables, or variables in denominators.
• Thus, solving them often involves basic arithmetic operations.

Getting familiar with the structure of these equations will make it easier to pick the suitable steps needed to isolate and solve for our desired variable.

Variable Isolation

Variable isolation involves performing operations to get a specific variable alone on one side of the equation. This is crucial in finding an expression for one variable in terms of others.Consider our equation: \[ 5x + 2y = 0 \]1. **Identify the term to isolate:** Here, \( y \) is the focus. We need to isolate it on one side of the equation.2. **Move other terms:** To do this, subtract \( 5x \) from both sides. This step is essential as it clears \( y \) of any additional terms, leading us to:\[ 2y = -5x \]3. **Final isolation:** Even though \( 5x \) is moved, \( y \) is still multiplied by 2. Divide all terms by 2 to solve for \( y \):\[ y = -\frac{5}{2}x \]Successful variable isolation simplifies analysis and plotting of relationships within the equation, providing clear insights into how variables depend on one another.

Algebraic Manipulation

Algebraic manipulation refers to the processes used to change and rearrange equations, making them easier to solve or analyze. In our exercise, algebraic manipulation involves operations like addition, subtraction, multiplication, and division.

• **Addition/Subtraction:** These operations help to move terms from one side of the equation to the other.
• **Multiplication/Division:** When isolating a variable, if that variable is multiplied by a coefficient, division is used to simplify it to 1.

In the equation \( 5x + 2y = 0 \):- To move \( 5x \), we perform subtraction because it's initially added. This operation transforms the equation to \( 2y = -5x \), effectively cleansing the side you want to be simplified.- Finally, dividing by 2 (since \( y \) is multiplied by it) simplifies the variable to a coefficient of 1, completing the manipulation: \[ y = -\frac{5}{2}x \]Proficiency in manipulating equations allows for a seamless transition and greater ease when faced with more complex algebraic expressions in future problems.