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Chapter 1: Problem 8

Determine whether the equation defines \(y\) as a linear function of \(x .\) If so, write it in the form \(y=m x+b\). $$ 3 \sqrt{x}+4 y=0 $$

Short Answer

Expert verified
The given equation is \(y = -\frac{3}{4}\sqrt{x}\). This does not follow the linear function form \(y = mx + b\), as it has a \(\sqrt{x}\) term instead of a constant multiplied by x. Consequently, the equation does not define y as a linear function of x.

Step by step solution

01

Isolate y in the equation

To begin, let's isolate y in the given equation: $$ 3 \sqrt{x} + 4y = 0 $$ Subtract \(3\sqrt{x}\) from both sides of the equation: $$ 4y = -3\sqrt{x} $$ Now, divide both sides by 4: $$ y = -\frac{3}{4} \sqrt{x} $$
02

Determine if the equation is a linear function

For a function to be linear in x, it must follow the form: $$ y = mx + b $$ Where m and b are constants. In our case, the equation we have is: $$ y = -\frac{3}{4} \sqrt{x} $$ We can see that it does not follow the linear function form, as the term \(\sqrt{x}\) is present instead of a constant multiplied by x. Thus, the given equation does not define y as a linear function of x.

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Key Concepts

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

Isolating Variables
Understanding how to isolate variables is crucial when solving equations. Isolating a variable means rearranging an equation so that one variable stands alone on one side of the equation, with all other terms on the opposite side. This simplification process aims to find the value of the variable.

To isolate a variable, one must perform operations such as addition, subtraction, multiplication, division, and even taking roots, provided these are done inversely to maintain the equation's balance. Let鈥檚 consider the initial step of the provided solution: To isolate the variable 'y', we subtracted \(3\sqrt{x}\) from both sides of the equation \(3\sqrt{x} + 4y = 0\), yielding \(4y = -3\sqrt{x}\). Furthermore, we divided each term by 4 to fully isolate 'y', giving us \(y = -\frac{3}{4}\sqrt{x}\). By isolating 'y', we prepare the equation for further analysis, such as determining if it defines a linear function.
Square Roots
Square roots are a fundamental concept in mathematics, integral when working with quadratic equations and solving for a variable. The square root of a number x is a value that, when multiplied by itself, returns x.

In our equation, the square root appears as \(\sqrt{x}\), indicating the number that squared equals x. Manipulating square roots can be tricky since they do not behave like linear terms. Operations involving square roots must be handled with care to avoid errors. For instance, the presence of the square root in our equation \(y = -\frac{3}{4}\sqrt{x}\) is the main reason it cannot be classified as a linear function, which would require 'x' to be in the first power, not under a square root.
Function Properties
Function properties are characteristics that help define and differentiate types of functions. Linear functions, one of the most basic forms, have specific properties: they show a constant rate of change and graph as straight lines. The standard form of a linear function is \(y = mx + b\), with 'm' representing the slope and 'b' the y-intercept.

In contrast, the given equation \(y = -\frac{3}{4}\sqrt{x}\) does not exhibit these properties. Instead of a constant rate of change, the rate varies with the value of x due to the square root, and graphing this function would not result in a straight line but rather a curve. Hence, it is inappropriate to call this equation a linear function of x, highlighting the importance of understanding function properties when classifying equations.

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Most popular questions from this chapter

For each supply equation, where \(x\) is the quantity supplied in units of 1000 and \(p\) is the unit price in dollars, (a) sketch the supply curve and (b) determine the number of units of the commodity the supplier will make available in the market at the given unit price. $$ \frac{1}{2} x-\frac{2}{3} p+12=0 ; p=24 $$

For each demand equation, where \(x\) represents the quantity demanded in units of 1000 and \(p\) is the unit price in dollars, (a) sketch the demand curve and (b) determine the quantity demanded corresponding to the given unit price \(p\). $$ 5 p+4 x-80=0 ; p=10 $$

The Venus Health Club for Women provides its members with the following table, which gives the average desirable weight (in pounds) for women of a given height (in inches): $$ \begin{array}{lrrrrr} \hline \text { Height, } \boldsymbol{x} & 60 & 63 & 66 & 69 & 72 \\ \hline \text { Weight, } \boldsymbol{y} & 108 & 118 & 129 & 140 & 152 \\ \hline \end{array} $$ a. Plot the weight \((y)\) versus the height \((x)\). b. Draw a straight line \(L\) through the points corresponding to heights of \(5 \mathrm{ft}\) and \(6 \mathrm{ft}\). c. Derive an equation of the line \(L\). d. Using the equation of part (c), estimate the average desirable weight for a woman who is \(5 \mathrm{ft}, 5\) in. tall.

Show that two distinct lines with equations \(a_{1} x+b_{1} y+\) \(c_{1}=0\) and \(a_{2} x+b_{2} y+c_{2}=0\), respectively, are parallel if and only if \(a_{1} b_{2}-b_{1} a_{2}=0\). Hint: Write each equation in the slope-intercept form and compare.

Is there a difference between the statements "The slope of a straight line is zero" and "The slope of a straight line does not exist (is not defined)"? Explain your answer.
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