Problem 53 - Algebra 1: Concepts and Skills

Question

Rewrite the equation so that \(y\) is a function of \(x\). $$x+\frac{2}{5} y=-1$$

Step-by-Step Solution

Answer

The equation \(y = -\frac{5}{2}x - \frac{5}{2}\) is \(y\) as a function of \(x\).

Step 1: Move \(x\) to the other side of the equation

To isolate \(y\), the terms without \(y\) need to be moved to the right side of the equation. This can be done by subtracting \(x\) from both sides of the equation. This results in the equation \(\frac{2}{5} y = -x -1\).

Step 2: Solve for \(y\)

Next, multiply both sides of the equation by the reciprocal of \(\frac{2}{5}\) which is \(\frac{5}{2}\). This will isolate \(y\). The result is \(y = -\frac{5}{2}x - \frac{5}{2}\).

Key Concepts

Functions

In mathematics, a function describes the relationship between two variables, typically called the input and the output. For a given input value, a function assigns exactly one output value. This unique pairing is what makes functions so vital in math and various scientific applications.

Functions are often described using equations where one variable depends on another. In our context, the goal is to express the variable \(y\) as a function of \(x\). In this setup, \(x\) acts as the independent variable, while \(y\) is the dependent variable, meaning its value changes based on \(x\).

Understanding this dependency helps us analyze how changes in \(x\) affect changes in \(y\). By writing \(y\) as a function of \(x\), we create a simple equation to predict \(y\) for any given \(x\). This process is crucial for solving mathematical problems and understanding relationships in everything from physics to economics.

Solving for y

Solving for \(y\) in an equation means rewriting the equation so that \(y\) is alone on one side, usually the left side. This allows you to easily calculate \(y\)'s value for any given \(x\).

Let's look at the problem where the initial equation is \(x + \frac{2}{5} y = -1\). To solve for \(y\), you must perform a series of algebraic steps.

• First, move terms involving \(x\) to the opposite side by subtracting \(x\) from both sides. This is shown as \(\frac{2}{5} y = -x - 1\).
• Next, isolate \(y\) by eliminating the coefficient \(\frac{2}{5}\). To do this, multiply both sides of the equation by the reciprocal \(\frac{5}{2}\), resulting in \(y = -\frac{5}{2}x - \frac{5}{2}\).

Through these steps, \(y\) is expressed explicitly in terms of \(x\). This makes it clear how \(y\) changes when \(x\) changes, enabling further analysis and calculation.

Isolating Variables

Isolating variables is an essential algebraic skill that involves rearranging an equation so that one particular variable stands alone on one side of the equation. This process often requires several steps of manipulation using operations such as addition, subtraction, multiplication, and division.

The purpose of isolating a variable, such as \(y\), is to see clearly how it is affected by other variables or constants in the equation. Let's apply this to isolate \(y\) starting from \(x + \frac{2}{5} y = -1\).

• We first subtract \(x\) from both sides to begin isolating \(y\), leading to \(\frac{2}{5} y = -x - 1\).
• To further isolate \(y\), multiply through by the reciprocal of the coefficient of \(y\), which is \(\frac{5}{2}\), giving \(y = -\frac{5}{2}x - \frac{5}{2}\).

This systematic isolation is the foundation for solving equations, allowing us to find precise values for the variable in question. Mastering this skill is crucial for tackling increasingly complex mathematical problems effectively.