Problem 77 - Introductory Algebra for College Students

Question

Exercises \(75-77\) will help you prepare for the material covered in the next section. Solve for \(y: 2 x+5 y=0\)

Step-by-Step Solution

Answer

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

Step 1: Isolate y-term

The aim is to make \(y\) the subject of the formula. Start by subtract \(2x\) from both sides of the equation: \(2x + 5y - 2x = 0 - 2x\). The \(2x\) terms on the left side of the equation cancel out, leaving \(5y = -2x\)

Step 2: Solve for y

Next, divide each side of the equation by the coefficient of \(y\) which is \(5\): \(\frac{5y}{5} = \frac{-2x}{5}\). The \(5\)'s on the left side of the equation cancel each other out, giving the final equation \(y = -\frac{2}{5}x\)

Key Concepts

Isolating the Variable

In linear equations, isolating the variable is a crucial step towards solving equations. When we want to solve an equation for a particular variable, such as in the given problem where we need to solve for \( y \), we must first get all the terms involving this variable on one side of the equation. This process makes one side of the equation depend solely on the variable we are solving for.
\[ 2x + 5y = 0 \]
The very first thing we do here is subtract \( 2x \) from both sides. This has the effect of cancelling out the \( 2x \) on the left side of the equation. Subtraction is a basic inverse operation that helps in removing unwanted terms, thus isolating the chosen variable's term. After this step, the equation evolves to:
\[ 5y = -2x \]
This is the essential stage of isolating the variable, which paves the way to easily solve for \( y \).

Solving for y

After isolating the \( y \) term, the next step is to solve for \( y \) by performing operations that eliminate any other coefficients associated with it. In the equation \( 5y = -2x \), the number \( 5 \) is a coefficient paired with \( y \). To simplify the equation simply to \( y \), we need to divide each term by this coefficient. Division is used here because it effectively reduces the coefficient of \( y \) to \( 1 \), leaving just \( y \) on one side.
\[ \frac{5y}{5} = \frac{-2x}{5} \]
By dividing \( 5y \) by \( 5 \), we remove the coefficient of \( y \) and achieve \( y = -\frac{2}{5}x \). Now \( y \) is isolated, and the equation clearly shows its relationship with \( x \). Understanding the correct manipulation of coefficients is key to solving for any variable in linear equations.

Equation Manipulation

To solve equations, sometimes we need to alter them while maintaining their equality. This process is known as equation manipulation. Each time we perform an operation on one side of an equation, we must do the same to the other side to keep things balanced.
In our particular example, the manipulation included subtracting \( 2x \) and dividing by \( 5 \). These operations are not arbitrary; they are chosen based on their ability to bring us closer to a simplified version of the equation where the desired variable is easily accessible.
The manipulation strategy typically involves:

• Adding or subtracting terms to both sides to eliminate parts of the equation or shift terms from one side to another.
• Multiplying or dividing through by constants to simplify or reduce the equation to a simpler form.

Practicing these techniques is fundamental not only for solving basic algebra problems but also for tackling more complex equations later on. This skill helps to better understand the structure of equations, as alterations allow us to view the relationships between different mathematical expressions.