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Chapter 2: Problem 62

Determine whether the equation defines \(y\) as a function of \(x.\) $$3 x+7 y=21$$

Short Answer

Expert verified
Yes, the equation defines \( y \) as a function of \( x \).

Step by step solution

01

Isolate the Variable

To determine if the equation defines \( y \) as a function of \( x \), we need to solve for \( y \) in terms of \( x \). Start by isolating \( y \) on one side of the equation.Given: \( 3x + 7y = 21 \) Subtract \( 3x \) from both sides: \[ 7y = 21 - 3x \]
02

Solve for y

Now, divide both sides of the equation by 7 to solve for \( y \).\[ y = \frac{21 - 3x}{7} \] Simplify the right-hand side:\[ y = 3 - \frac{3}{7}x \]
03

Define y as a Function of x

The equation \( y = 3 - \frac{3}{7}x \) is in the format \( y = f(x) \), which clearly shows that \( y \) is expressed in terms of \( x \). Since there is only one value of \( y \) for each value of \( x \), the equation defines \( y \) as a function of \( x \).

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Key Concepts

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

Understanding Linear Equations
A linear equation is a type of equation where the highest power of the variable is 1. This means the graph of a linear equation is always a straight line. Such equations can be written in the general form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. In our exercise, the equation \( 3x + 7y = 21 \) is a linear equation.

Characteristics of a linear equation include:
  • It involves no exponents or powers other than 1.
  • The variables are always first-degree.
  • The graph always forms a straight line on a coordinate plane.
Understanding that our given equation is linear helps simplify the process of analyzing and solving it. It also informs us that the behavior of \( y \) with respect to \( x \) will be consistent and predictable, which is essential when determining functions.
Steps to Solving for y
Solving for \( y \) in an equation involves isolating \( y \) on one side of the equation, resulting in an expression that shows how \( y \) depends on \( x \). This process can be understood better with the help of a sequence of steps.
  • Step 1: Isolate the terms with \( y \). In the equation \( 3x + 7y = 21 \), we subtract \( 3x \) from both sides to isolate terms with \( y \). This gives us \( 7y = 21 - 3x \).
  • Step 2: Solve for \( y \). Next, divide both sides by 7 to fully isolate \( y \). This results in \( y = \frac{21 - 3x}{7} \).
  • Simplify the expression. Simplifying gives us \( y = 3 - \frac{3}{7}x \).
Following these steps allows us to express \( y \) in terms of \( x \), helping us identify relationships between the two variables in a clear format. This is crucial when checking if \( y \) can be defined as a function of \( x \).
Determining Functions
A function is a relationship where each input has a single output. In mathematical terms, this means that a function \( y = f(x) \) associates every value of \( x \) with only one value of \( y \).

For the equation \( y = 3 - \frac{3}{7}x \), we can say that \( y \) is a function of \( x \) because:
  • The expression \( y = 3 - \frac{3}{7}x \) clearly defines one value of \( y \) for every value of \( x \).
  • There are no operations or cube roots that might produce multiple values of \( y \) for a single value of \( x \).
  • We have a straight line graph which is characteristic of linear functions, verifying it's indeed a function.
By understanding these characteristics, you can confidently determine whether an equation like \( 3x + 7y = 21 \) specifies \( y \) as a function of \( x \). This clarity facilitates deeper comprehension and application of linear equations and functions in various mathematical contexts.

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Most popular questions from this chapter

Step Functions In Example 8 and Exercises 89 and 90 we are given functions whose graphs consist of horizontal line segments. Such functions are often called step functions, because their graphs look like stairs. Give some other examples of step functions that arise in everyday life.

A tank holds 100 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 40 minutes. Toricelli's Law gives the volume of water remaining in the tank after \(t\) minutes as $$V(t)=100\left(1-\frac{t}{40}\right)^{2}$$ (a) Find \(V^{-1} .\) What does \(V^{-1}\) represent? (b) Find \(V^{-1}(15) .\) What does your answer represent?

The population \(P\) (in thousands) of San Jose, California, from 1988 to 2000 is shown in the table. (Midyear estimates are given.) Draw a rough graph of \(P\) as a function of time \(t\). $$\begin{array}{|c|c|}\hline t & P \\\\\hline 1988 & 733 \\\1990 & 782 \\\1992 & 800 \\\1994 & 817 \\\1996 & 838 \\\1998 & 861 \\\2000 & 895 \\\\\hline\end{array}$$

In Exercise 65 of Section 2.7 you were asked to solve equations in which the unknowns were functions. Now that we know about inverses and the identity function (see Exercise \(82),\) we can use algebra to solve such equations. For instance, to solve \(f \circ g=h\) for the unknown function \(f\) we perform the following steps: \(f \circ g=h\) Problem: Solve for \(f\) \(f \circ g \circ g^{-1}=h \circ g^{-1} \quad\) Compose with \(g^{-1}\) on the right \(f \circ I=h \circ g^{-1} \quad g \circ g^{-1}=I\) \(f=h \circ g^{-1} \quad\) f \(\circ I=f\) So the solution is \(f=h \circ g^{-1} .\) Use this technique to solve the equation \(f \circ g=h\) for the indicated unknown function. (a) Solve for \(f,\) where \(g(x)=2 x+1\) and \(h(x)=4 x^{2}+4 x+7\) (b) Solve for \(g,\) where \(f(x)=3 x+5\) and \(h(x)=3 x^{2}+3 x+2\)

Bird Flight \(\quad\) A bird is released from point \(A\) on an island, \(5 \mathrm{mi}\) from the nearest point \(B\) on a straight shoreline. The bird flies to a point \(C\) on the shoreline, and then flies along the shoreline to its nesting area \(D\) (see the figure). Suppose the bird requires \(10 \mathrm{kcal} / \mathrm{mi}\) of energy to fly over land and \(14 \mathrm{kcal} / \mathrm{mi}\) to \(\mathrm{fly}\) over water (see Example 9 in Section 1.6 ). (a) Find a function that models the energy expenditure of the bird. (b) If the bird instinctively chooses a path that minimizes its energy expenditure, to what point does it fly? (cant copy image)
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