Problem 16 - Algebra 1

Question

Rewrite the equation so that \(y\) is a function of \(x .\) $$3 x+5 y=7$$

Step-by-Step Solution

Answer

The equation, with y expressed as a function of x, is y = (7 - 3x)/5

Step 1: Isolate y

To isolate y, the first step is to move the term involving x to the other side by subtracting 3x from both sides. Doing so, the equation becomes: 5y = 7 - 3x

Step 2: Solve for y

The next step involves further modifying the equation so that y stands alone on one side. To get y, divide both sides of the equation by 5, which gives: y = (7 - 3x)/5

Key Concepts

Solving Equations

Solving equations is a fundamental part of algebra, especially when working with linear equations. A linear equation is typically in the form of \( a x + b y = c \), where \( a \), \( b \), and \( c \) are constants. Our goal is to find the value of the variable, in this case, \( y \), that makes the equation true.
To solve the equation \( 3x + 5y = 7 \), we need to manipulate it to express \( y \) solely in terms of \( x \). This process includes strategically performing operations like addition, subtraction, multiplication, or division on both sides to keep the equation balanced.

• Isolate: Begin by moving terms from one side of the equation to the other. This often involves subtraction or addition, as seen when we subtract \( 3x \) from both sides to isolate terms involving \( y \).
• Balance: Remember, what you do to one side of the equation, you must do to the other. This ensures that the equation stays true!
• Divide: Once you have isolated \( y \, \), divide by its coefficient to solve for \( y \) completely.

By following these steps, you turn a complex equation into a simpler, more understandable expression.

Functions

Understanding functions is crucial when dealing with equations where one variable is expressed in terms of another. A function defines a particular relationship between two sets of data, typically \( x \) (the independent variable) and \( y \) (the dependent variable). In the equation \( y = \frac{7 - 3x}{5} \), \( y \) is expressed as a function of \( x \).

• Independent vs. Dependent: Here, \( x \) is independent, meaning you can choose its value freely. \( y \), the dependent variable, is determined by what value \( x \) takes.
• Notation: Functions are often denoted as \( f(x) \), where \( f \) represents the function itself. In some cases, you could rewrite \( y = \frac{7 - 3x}{5} \) as \( f(x) = \frac{7 - 3x}{5} \).
• Graphs: Functions can be graphed on a coordinate plane, where the \( x \)-axis represents the independent variable and the \( y \)-axis presents the dependent.

Grasping the concept of functions helps in seeing the relationship between the two variables, depicting it visually, and understanding how changes in one variable affect the other.

Algebraic Manipulation

Algebraic manipulation involves changing the form of an equation or expression to make it easier to work with or solve. This is a skill that requires practice and understanding of basic algebraic operations. In our example, algebraic manipulation allowed us to transform \( 3x + 5y = 7 \) into \( y = \frac{7 - 3x}{5} \).

• Use of Operations: It includes using operations like addition, subtraction, multiplication, and division to simplify or rearrange equations.
• Factoring and Expanding: These techniques may also be involved, although not required in this specific problem.
• Combining Like Terms: Simplifying expressions by combining terms with the same variable part is a core component of manipulation.

Effective algebraic manipulation allows us to uncover solutions more straightforwardly and helps in creating a more understandable expression. It’s about working methodically to simplify complex equations, making them look more familiar and easier to solve. Working with this tool helps build a strong algebraic foundation.