Problem 9 - Algebra 1

Question

Rewrite the equation \(2 x+2 y=10\) so that \(y\) is a function of \(x .\)

Step-by-Step Solution

Answer

The equation \(2x + 2y = 10\) rewritten to express \(y\) as a function of \(x\) is \(y = 5 - x\).

Step 1: Subtract \(2x\) from both sides of the equation

We subtract \(2x\) from both sides of the equation to start isolating \(y\). This gives us \(2y = 10 - 2x\).

Step 2: Divide by 2

Next, we divide every term by 2 in order to isolate \(y\). After division, the equation becomes \(y = 5 - x\).

Step 3: Present the final form of the equation

The equation now have \(y\) isolated and it is expressed in terms of \(x\). Therefore, the function of \(x\) is \(y = 5 - x\).

Key Concepts

Function Notation

Function notation is a way to express one variable as a dependent function of another variable. Instead of writing equations using traditional symbols, function notation uses format like \( y = f(x) \), where \( f(x) \) signifies a particular expression with \( x \) as the input.

• In function notation, \( y = f(x) \) indicates that \( y \) is the output for a given input \( x \).
• We often use function notation to describe relationships where one variable directly depends on another, such as in mathematical models or real-world scenarios.

In the original exercise, we had \(2x + 2y = 10\), and we transformed it into the function notation \(y = 5 - x\), making \(y\) explicitly dependent on \(x\). This notation makes it easier to evaluate and analyze how changes in \(x\) affect \(y\), as well as to graph the relationship.

Solving Equations

Solving equations involves finding the value of the unknown variable(s) that makes an equation true. When equations involve linear terms, like in the given exercise, the solution aims to express this relationship in a simplified form.

• The step-by-step solution involved two main actions: subtracting \(2x\) and then dividing by 2. These operations gradually assist in simplifying the left side of the equation to only have \(y\).
• Subtraction helps reduce the complexity of the equation, shifting terms to one side, while division assists in normalizing coefficients.

Understanding the order of operations is crucial in solving equations. It involves strategically removing or simplifying terms to isolate variables effectively. The practice of subtraction and division follows the **inverse operations principle**, which is key in solving linear equations.

Isolation of Variables

Isolation of variables refers to the process of manipulating an equation so that one variable stands alone on one side of the equation. This technique is used to express the variable in terms of other variables or constants. In algebra, isolation is a fundamental step in solving equations.

• To isolate a variable, utilize inverse operations such as addition, subtraction, multiplication, or division.
• In the original exercise, \(y\) was isolated to make it a function of \(x\), resulting in the equation \(y = 5 - x\).

The process involves moving terms step-by-step:1. Subtracting \(2x\) from both sides removed the term involving \(x\) from the side of \(2y\).2. Dividing by 2 allowed \(y\) to be on its own, representing its relationship directly with \(x\).By isolating variables, we can easily understand dependencies and potentially solve further problems, infer conditions, or graph the relationships for deeper analysis. This enhances our mathematical problem-solving skills by enabling more straightforward manipulations of equations.