Problem 23 - Algebra 1: Concepts and Skills

Question

Rewrite the equation in function form. $$ 2 x+3 y=6 $$

Step-by-Step Solution

Answer

The equation in function form is \(y = 2 - \frac{2}{3}x\).

Step 1: Isolate y

Subtract \(2x\) from both sides of the given equation \(2x + 3y = 6\). This will give the equation \(3y = 6 - 2x\).

Step 2: Solve for y

Now, divide both sides by \(3\). This gives the equation in function form: \(y = 2 - \frac{2}{3}x\).

Key Concepts

Equation Solving

Equation solving is a fundamental skill in mathematics that involves finding the value of the unknown variable that makes the equation true. When we solve equations, especially in algebra, our goal is to simplify complex expressions and isolate the variable we need to solve for.

• The first step in solving equations is often to perform arithmetic operations such as addition, subtraction, multiplication, or division to both sides of the equation.
• This helps to simplify the equation and lays the foundation for isolating the variable.

In the problem provided, the initial task is to rewrite an equation in function form, which is a common requirement in algebra. To accomplish this, familiarity with linear equations and the ability to effectively manipulate and solve them is crucial. Once the equation is in a simpler form, the variable of interest, usually represented as 'y', can be expressed clearly in terms of other variables and constants.

Isolating Variables

Isolating a variable means rearranging an equation so that one variable stands alone on one side of the equals sign. This process is essential in expressing relationships in mathematical terms and exploring how changes in one variable affect another. To isolate a variable, follow these steps:

• Perform inverse operations: If a variable is added, subtract it; if it is multiplied, divide it.
• Step by step, undo operations to keep the equation balanced.
• Always perform the same operation on both sides of the equation to maintain equality.

In our exercise, we isolate the variable 'y'. Beginning with the equation \(2x + 3y = 6\), we subtract \(2x\) from both sides, resulting in \(3y = 6 - 2x\). Finally, dividing both sides by 3 simplifies the equation to \(y = 2 - \frac{2}{3}x\). This isolating process helps present the function in a clear, understandable form.

Linear Equations

Linear equations are equations of the first degree, which means their highest exponent of the variable is one. These equations form straight lines when graphed and are expressed in the standard format \(ax + by = c\). Understanding linear equations is crucial in several disciplines, such as physics and economics, where they model relationships between different variables.

• They often appear in the form \(y = mx + b\), which is known as the slope-intercept form.
• 'm' represents the slope of the line, and 'b' is the y-intercept, where the line crosses the y-axis.

By converting our equation \(2x + 3y = 6\) to function form \(y = 2 - \frac{2}{3}x\), we transform it into a linear format that is easy to analyze. The function form is useful for quickly identifying the slope and intercept, allowing for better insight into how changes in 'x' affect 'y'. Whether used for simple equations or complex modeling, grasping linear equations enhances problem-solving capabilities in mathematics.