Problem 72 - Elementary Algebra

Question

Solve for \(y: 5 x+2 y=3\)

Step-by-Step Solution

Answer

The solution is \( y = \frac{3 - 5x}{2} \).

Step 1: Identify the Equation

We are given the equation \(5x + 2y = 3\). Our goal is to solve for \(y\).

Step 2: Isolate the Term Containing y

Subtract \(5x\) from both sides of the equation to isolate the term with \(y\) on one side:\[ 2y = 3 - 5x \]

Step 3: Solve for y

Divide every term by 2 to solve for \(y\):\[ y = \frac{3 - 5x}{2} \]

Key Concepts

Isolation of Variables

In solving linear equations, the concept of isolation of variables is essential. It's about moving the variable of interest to one side of the equation, allowing us to express it solely in terms of other quantities.
For example, consider the original exercise equation: \[ 5x + 2y = 3 \]To solve for \(y\), we need to isolate \(y\) on one side. To do this, we subtract \(5x\) from both sides, which results in: \[ 2y = 3 - 5x \] This operation ensures that all terms involving \(y\) are on one side and everything else is on the other. Building up such a routine helps solve more complex equations.
Note the importance of performing the same operation on both sides of the equation to maintain the equality. This step is crucial in linear algebra, emphasizing the balance that must be preserved in equations.

Algebraic Manipulation

Algebraic manipulation is a fundamental tool in solving equations. It involves rearranging and simplifying equations to get a variable in its desired form. After isolating the variable containing \(y\), the next step involves simplifying this expression.
Looking at our equation:\[ 2y = 3 - 5x \]The goal here is to "free" \(y\) from any multiplying constants. We achieve this by dividing each term by the number next to \(y\), which in this case is 2. Performing this operation:\[ y = \frac{3 - 5x}{2} \] This ensures that \(y\) is now alone on its side of the equation, expressed clearly in terms of \(x\).
These steps of multiplication and division are key aspects of algebraic manipulation, allowing one to simplify equations and solve for variables effectively.

Two-Variable Equations

Equations involving two variables, like \(x\) and \(y\), are common in algebra. Understanding them is vital because they represent relationships between quantities. For example, our equation:\[ 5x + 2y = 3 \] presents a linear relationship between \(x\) and \(y\). Such equations often model real-world situations where two quantities depend on one another.
When dealing with two-variable equations, you often need to decide which variable to solve for based on context.
Solving these equations can result in solutions that define one variable in terms of the other. Particularly in this case, solving for \(y\) yields the solution: \[ y = \frac{3 - 5x}{2} \], which tells us that for any given value of \(x\), there's a corresponding value of \(y\).
Mastery of these concepts is crucial for progressing in mathematics, as it sets the foundation for solving systems of equations and understanding functions.