Problem 15 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The equation rewritten with \(y\) as a function of \(x\) is \(y = 5 - 2x\).

Step 1: Analyze the equation

The given equation is \(2x + y = 5 \). The goal is to express y as a function of x.

Step 2: Isolate y term

For this, the term containing x (which is 2x) should be subtracted from both sides of the equation. This gives: \(y = 5 - 2x\)

Step 3: Verification

To verify this result, substitute \(y = 5 - 2x\) into the original equation and see if both sides are equivalent. This yields \(2x + 5 - 2x = 5\), which simplifies to \(5 = 5 \) thus verifying the result.

Key Concepts

Functions of x

In mathematics, when we say that something is a "function of \( x \)," we are looking to express one variable explicitly in terms of another. In this context, expressing a relationship as a function of \( x \) allows us to easily see how changes in \( x \) will affect the other variable.

For example, take our equation \( 2x + y = 5 \). We want to express \( y \) solely in terms of \( x \). This means rearranging the equation to solve for \( y \), giving us a clear function \( y(x) \). In this case, the function of \( x \) would be \( y = 5 - 2x \).

This expression shows a direct relationship between \( x \) and \( y \). For every unit increase in \( x \), \( y \) decreases by 2 units, reflecting a linear relationship. Understanding functions of \( x \) helps in analyzing and predicting the outcome of changes in \( x \) on the function's value.

• Functions convey relationships between two quantities.
• Useful for understanding dependencies.
• Essential for graphing and analyzing equations.

Expressing Variables

Expressing variables involves rearranging equations to isolate a specific variable. This process is crucial for uncovering the relationships between different variables in an equation. In our exercise, we begin with \( 2x + y = 5 \), where both \( y \) and \( x \) are mixed together.

To clarify the dependency of \( y \) on \( x \), it's necessary to express \( y \) independently. This means moving all other terms to the opposite side of the equation. We can achieve this for the equation by subtracting \( 2x \) from both sides, resulting in \( y = 5 - 2x \).

When expressing one variable in terms of another, it's important to:

• Identify which variable you want to isolate.
• Use inverse operations to move other terms across the equation.
• Ensure the isolated variable is on its own on one side of the equation.

Expressing variables clearly allows us to understand how one variable affects another.

Solving Equations

Solving equations is about finding values of variables that make the equation true. In a linear equation like \( 2x + y = 5 \), solving for a particular variable involves simplifying the equation step-by-step until the desired variable is isolated.

The process often includes:

• Identifying terms containing the variable.
• Rearranging terms using addition, subtraction, multiplication, or division.
• Simplifying the expression to maintain the equation's equality.

In our example, we solved for \( y \) to get \( y = 5 - 2x \), which shows \( y \) in terms of \( x \).

Verification is a crucial part of solving equations. After solving an equation, substitute the result back into the original equation to confirm correctness. For instance, substituting \( y = 5 - 2x \) back into \( 2x + y = 5 \) verifies that both sides equal 5, ensuring the solution is correct.

Solving equations is a foundational skill in mathematics, crucial for understanding and analyzing mathematical relationships.