Problem 23 - Algebra 1

Question

Rewrite the equation so that \(y\) is a function of \(x .\) $$-3 x+4 y-5=-14$$

Step-by-Step Solution

Answer

The function of 'y' in terms of 'x' is \(y = \frac{3x + 9}{4}\)

Step 1: Identify the equation

The given equation is \(-3x + 4y - 5 = -14\)

Step 2: Isolate '4y' term

Get '4y' alone on one side by adding '3x' and '5' to both sides. This yields: \(4y = 3x - 5 + 14 = 3x + 9\)

Step 3: Solve for 'y'

Finally, divide by 4 on both sides to get 'y' alone. This gives: \(y = \frac{3x + 9}{4}\)

Key Concepts

Isolating Variables

Isolating variables is a fundamental aspect of solving linear equations. It involves using arithmetic operations to get the variable of interest, such as 'y' in our example, alone on one side of the equation. This is a key step because it converts equations into a more usable form, particularly a function, where we express one variable directly in terms of another.

For instance, in the equation \( -3x + 4y - 5 = -14 \), our goal is to isolate \( y \). To do this, we need to 'move' the other terms involving \( x \) to the opposite side. Remember, the golden rule is to do the same operation on both sides to maintain the equation's balance. By adding \( 3x \) and \( 5 \) to both sides, we’re effectively shifting all non-\( y \) terms away, setting the stage for \( y \) to stand alone.

Once we have \( y \) by itself on one side, we have isolated the variable successfully, and the equation is one step closer to being in function form, which is very useful for various mathematical and real-world applications.

Equation Manipulation

Equation manipulation is much like a balancing act. The main objective when dealing with linear equations is to perform operations that simplify or rearrange the equation to help us solve for a variable. In the context of our equation, we manipulate it by performing arithmetic operations: addition, subtraction, multiplication, or division.

The manipulation starts by adding \( 3x \) and \( 5 \) to both sides of our starting equation \( -3x + 4y - 5 = -14 \). These steps give us \( 4y = 3x + 9 \). We then encounter another manipulation step – division. To get \( y \) by itself, we divide everything by the coefficient of \( y \) which is 4, arriving at the much simpler \( y = \frac{3x + 9}{4} \). Each operation is performed with precision to ensure the equation's integrity remains intact while reaching the solution. This skill is not only essential for solving equations but is also a cornerstone for more advanced mathematical concepts.

Functions of X

In algebra, a function of \( x \) expresses the relationship between two variables — usually \( x \) and \( y \). We say \( y \) is a function of \( x \) if for every value of \( x \) there is exactly one value of \( y \). When we isolate one variable in terms of another, the equation essentially becomes a function.

In the given equation \( y = \frac{3x + 9}{4} \), \( y \) is now expressed as a function of \( x \) because for each input of \( x \) there is a corresponding output of \( y \) calculated through the equation. This is powerful because it provides a clear way to see how changes in \( x \) affect \( y \) — a fundamental aspect of understanding functions in mathematics.

Functions are used in various fields like economics, science, and engineering to model relationships and predict outcomes. Being able to rewrite an equation as a function of \( x \) enables students to graph the equation and make sense of the relationship visually, thereby deepening their understanding of these concepts.