Chapter 9

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies inversely as \(a^{4}\)

Solving systems involving more than three variables can be accomplished with methods similar to those encountered in this section. Apply what you already know to solve each system of equations in four variables. $$ \left\\{\begin{aligned} x+y &-w=0 \\ y+2 z+w &=3 \\ x-z &=1 \\ 2 x-y &-w=-1 \end{aligned}\right. $$

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies directly as \(x\) and inversely as \(p^{2}\)

Solving systems involving more than three variables can be accomplished with methods similar to those encountered in this section. Apply what you already know to solve each system of equations in four variables. $$ \left\\{\begin{aligned} 5 x+4 y &=29 \\ y+z-w &=-2 \\ 5 x+z &=23 \\ y-z+w &=4 \end{aligned}\right. $$

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies directly as \(a^{5}\) and inversely as \(b\)

Solving systems involving more than three variables can be accomplished with methods similar to those encountered in this section. Apply what you already know to solve each system of equations in four variables. $$ \left\\{\begin{aligned} x+y+z+w &=5 \\ 2 x+y+z+w &=6 \\ x+y+z &=2 \\ x+y &=0 \end{aligned}\right. $$

Solving systems involving more than three variables can be accomplished with methods similar to those encountered in this section. Apply what you already know to solve each system of equations in four variables. $$ \left\\{\begin{aligned} 2 x \quad-z &=-1 \\ y+z+w &=9 \\ y-2 w &=-6 \\ x+y &=3 \end{aligned}\right. $$

Write a system of three linear equations in three variables that are dependent equations.

The number of personal bankruptcy petitions filed in the United States was consistently on the rise until there was a major change in bankruptcy law. The year 2007 was the year in which the fewest personal bankruptcy petitions were filed in 15 years, but the rate soon began to rise. In \(2009,\) the number of petitions filed was 206,593 less than twice the number of petitions filed in 2007 . This is equivalent to an increase of 568,751 petitions filed from 2007 to 2009. Find how many personal bankruptcy petitions were filed in each year. (Source: Based on data from the Administrative Office of the United States Courts)

Find the slope of the line containing each pair of points. (3,6),(-2,6)

Find the slope of the line containing each pair of points. (-5,-2),(0,7)

Find the values of \(a, b\), and \(c\) such that the equation \(y=a x^{2}+b x+c\) has ordered pair solutions \((1,6),(-1,-2),\) and \((0,-1) .\) To do so, substitute each ordered pair solution into the equation. Each time, the result is an equation in three unknowns: \(a, b,\) and \(c .\) Then solve the resulting system of three linear equations in three unknowns, \(a, b,\) and \(c\).

Find the slope of the line containing each pair of points. (4,-1),(5,-2)

Find the values of \(a, b,\) and \(c\) such that the equation \(y=a x^{2}+b x+c\) has ordered pair solutions \((1,2),(2,3),\) and \((-1,6) .(\) Hint: See Exercise 61.)

Find the slope of the line containing each pair of points. (2,1),(2,-3)

Data \((x, y)\) for the total number (in thousands) of college-bound students who took the ACT assessment in the year \(x\) are approximately \((3,929),\) (11,1179) , and \((19,1470),\) where \(x=3\) represents 1993 and \(x=11\) represents 2001 . Find the values \(a\), \(b,\) and \(c\) such that the equation \(y=a x^{2}+b x+c\) models these data. According to your model, how many students will take the \(\mathrm{ACT}\) in \(2012 ?\) (Source: ACT, Inc.)

Solve. See the Concept Check in this section. Choose the type of variation that each equation represents. a. Direct variation b. Inverse variation c. Joint variation \(y=\frac{2}{3} x\)

Monthly normal rainfall data \((x, y)\) for Portland, Oregon, are \((4,2.47),(7,0.58),(8,1.07),\) where \(x\) represents time in months (with \(x=1\) representing January) and \(y\) represents rainfall in inches. Find the values of \(a, b,\) and \(c\) rounded to 2 decimal places such that the equation \(y=a x^{2}+b x+c\) models this data. According to your model, how much rain should Portland expect during September? (Source: National Climatic Data Center)

Solve. See the Concept Check in this section. Choose the type of variation that each equation represents. a. Direct variation b. Inverse variation c. Joint variation \(y=\frac{0.6}{x}\)

Solve. See the Concept Check in this section. Choose the type of variation that each equation represents. a. Direct variation b. Inverse variation c. Joint variation \(y=9 a b\)

Solve. See the Concept Check in this section. Choose the type of variation that each equation represents. a. Direct variation b. Inverse variation c. Joint variation \(x y=\frac{2}{11}\)

The volume of a cylinder varies jointly as the height and the square of the radius. If the height is halved and the radius is doubled, determine what happens to the volume.

The horsepower to drive a boat varies directly as the cube of the speed of the boat. If the speed of the boat is to double, determine the corresponding increase in horsepower required.

Suppose that \(y\) varies directly as \(x^{2}\). If \(x\) is doubled, what is the effect on \(y\) ?

Suppose that \(y\) varies directly as \(x .\) If \(x\) is doubled, what is the effect on \(y ?\)