Problem 52 - Algebra 1: Concepts and Skills

Question

Rewrite the equation so that \(y\) is a function of \(x\). $$-3 x+6 y=12$$

Step-by-Step Solution

Answer

The function of \(x\) is \(y = 0.5x + 2\).

Step 1: Isolate y term

Start with the original equation \(-3x + 6y = 12\). Add \(3x\) to both sides in order to isolate terms with \(y\) on one side. This leaves us with \(6y = 3x + 12\).

Step 2: Solve for y

Now, divide the entire equation by \(6\) to solve for \(y\). After division, we get \(y = 0.5x + 2\). This is the function of \(x\).

Key Concepts

Function of a Variable

In mathematics, a function of a variable describes a relationship between two variables, where one variable depends on the other. In the context of algebraic equations, when we express an equation as a function of a variable, it means solving the equation for one variable in terms of another. This allows us to see how changes in one variable affect the other.
For example, when we have an equation like \(y = 0.5x + 2\), it becomes clear that \(y\) is dependent on the value of \(x\).

• The function has the general form \(y = f(x)\), where \(f(x)\) represents an expression involving \(x\).
• Understanding the function makes it easy to predict the value of \(y\) for any given \(x\).

Functions are crucial in many areas of math because they simplify complex relationships and make it easier to analyze how variables interact.

Isolating a Variable

Isolating a variable is a fundamental algebraic technique used to express a specific variable by itself on one side of an equation. This process is essential when converting equations into functions, like solving for \(y\) in terms of \(x\) in a linear equation.
To isolate a variable, we perform operations that systematically simplify the equation.

• Start by moving all terms containing the variable you want to isolate to one side of the equation. This may involve adding or subtracting terms from both sides.
• Next, if the variable is multiplied by a number, divide both sides of the equation by that number to solve for the variable.

The ultimate aim is to have the isolated variable on one side, with all other terms on the opposite side. Isolating a variable helps in discovering functions and understanding relationships between different mathematical variables.

Linear Equations

Linear equations are a type of algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations can be expressed in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are real numbers.
These equations are called 'linear' because they graph as straight lines on a coordinate plane.

• Linear equations have a constant rate of change, which is represented by the slope. In the function \(y = 0.5x + 2\), the slope is \(0.5\).
• The equation also contains a constant term, which shifts the line up or down on the graph. Here, the constant is \(2\).

Solving linear equations involves finding the values of variables that make the equation true, often leading to a function description. Understanding linear equations is vital for interpreting data and predicting outcomes in various real-world applications.