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Chapter 2: Problem 19

For each formula, express y as a function of x. $$2 x+3 y=9$$

Short Answer

Expert verified
y = 3 - \frac{2}{3}x

Step by step solution

01

Isolate the term with y

To express y as a function of x, first, isolate the term that includes y. The given equation is \[\begin{equation} 2x + 3y = 9 onumber \ \text{Subtract } 2x \text{ from both sides:} onumber \ 3y = 9 - 2x onumber \end{equation}\]
02

Solve for y

Next, solve for y by dividing both sides of the equation by 3 to isolate y: \[\begin{equation} y = \frac{9 - 2x}{3} onumber \ \text{or} \ y = 3 - \frac{2}{3}x onumber \end{equation}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Isolating variables is a crucial step in solving equations.
The goal is to get the variable you're solving for by itself on one side of the equation.
For example, in the equation 2x + 3y = 9, we want to solve for y.
This involves eliminating the x term from the left side.
Start by subtracting 2x from both sides:
3y = 9 - 2x
By isolating y, we simplify our calculations and pave the way for solving the equation.
Always perform the same operation on both sides to keep the equation balanced.
Algebraic Equations
Algebraic equations involve variables and constants.
They are mathematical statements that assert the equality of two expressions.
In our example, the equation is 2x + 3y = 9.
The terms 2x and 3y are variable terms, while 9 is a constant.
Understanding the components of an equation is key to manipulating and solving it.
Equations can often be solved through a series of logical steps, such as isolating variables and simplifying expressions.
Solving Linear Equations
Linear equations are equations of the first degree, meaning they involve variables raised to the power of one.
The general form is ax + by = c.
Our example, 2x + 3y = 9, is a linear equation.
Solving linear equations typically involves isolating one variable and simplifying the equation.
For instance, after isolating y in 3y = 9 - 2x, we divide both sides by 3 to solve for y:
y = (9 - 2x)/3
This simplification turns the equation into a form that's easy to interpret and plot.
Mathematical Functions
A mathematical function describes a relationship between two variables.
When we express y as a function of x, we show how y changes with respect to x.
In our case, y = (9 - 2x)/3, we can clearly see that y depends on x.
The function form provides a straightforward way to understand and analyze this relationship.
By converting equations to functions, we can use them more effectively in various mathematical applications, such as graphing and modeling.

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