Chapter 9

Solve each system. $$ \left\\{\begin{array}{r} x+y+z=8 \\ 2 x-y-z=10 \\ x-2 y-3 z=22 \end{array}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{array}{r} y+2 x \geq 0 \\ 5 x-3 y \leq 12 \\ y \leq 2 \end{array}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{r} x+y+z=2 \\ 2 x-z=5 \\ 3 y+z=2 \end{array}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=100\) when \(x=7\)

Solve each system. $$ \left\\{\begin{aligned} 5 x+y+3 z &=1 \\ x-y+3 z &=-7 \\ -x+y &=1 \end{aligned}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{array}{rr} y+2 x \leq & 0 \\ 5 x+3 y \geq & -2 \\ y \leq & 4 \end{array}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{r} x+2 y+z=5 \\ x-y-z=3 \\ y+z=2 \end{array}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=63\) when \(x=3\)

Solve each system. $$ \left\\{\begin{array}{r} x+2 y-z=5 \\ 6 x+y+z=7 \\ 2 x+4 y-2 z=5 \end{array}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{aligned} 3 x-4 y & \geq-6 \\ 2 x+y \leq & 7 \\ y \geq &-3 \end{aligned}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{r} 5 x-2 y=27 \\ -3 x+5 y=18 \end{array}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=\frac{1}{8}\) when \(x=16\)

Solve each system. $$ \left\\{\begin{array}{r} 4 x-y+3 z=10 \\ x+y-z=5 \\ 8 x-2 y+6 z=10 \end{array}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{aligned} 4 x-y & \geq-2 \\ 2 x+3 y & \leq-8 \\ y & \geq-5 \end{aligned}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=\frac{1}{10}\) when \(x=40\)

Solve each system. $$ \left\\{\begin{array}{r} 2 x-3 y+z=2 \\ x-5 y+5 z=3 \\ 3 x+y-3 z=5 \end{array}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{aligned} 2 x+y & \leq 5 \\ x & \leq 3 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{aligned} 4 x-7 y &=7 \\ 12 x-21 y &=24 \end{aligned}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=0.2\) when \(x=0.7\)

Solve each system. $$ \left\\{\begin{array}{r} 4 x+y-z=8 \\ x-y+2 z=3 \\ 3 x-y+z=6 \end{array}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{aligned} 3 x+y &=4 \\ x & \leq 4 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{r} 2 x-5 y=12 \\ -4 x+10 y=20 \end{array}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=0.6\) when \(x=0.3\)

Solve each system. $$ \left\\{\begin{array}{rr} -2 x-4 y+6 z= & -8 \\ x+2 y-3 z= & 4 \\ 4 x+8 y-12 z= & 16 \end{array}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{rr} 4 x-y+2 z= & 5 \\ 2 y+z= & 4 \\ 4 x+y+3 z= & 10 \end{array}\right. $$

Pairs of markings a set distance apart are made on highways so that police can detect drivers exceeding the speed limit. Over a fixed distance, the speed \(R\) varies inversely with the time \(T\). In one particular pair of markings, \(R\) is 45 mph when \(T\) is 6 seconds. Find the speed of a car that travels the given distance in 5 seconds.

Solve each system. $$ \left\\{\begin{aligned} -6 x+12 y+3 z &=-6 \\ 2 x-4 y-z &=2 \\ -x+2 y+\frac{z}{2} &=-1 \end{aligned}\right. $$

The weight of an object on or above the surface of Earth varies inversely as the square of the distance between the object and Earth's center. If a person weighs 160 pounds on Earth's surface, find the individual's weight if he moves 200 miles above Earth. Round to the nearest whole pound. (Assume that Earth's radius is 4000 miles.)

Solve each system. $$ \left\\{\begin{aligned} 2 x+2 y-3 z &=1 \\ y+2 z &=-14 \\ 3 x-2 y &=-1 \end{aligned}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{aligned} 4 x+y+z &=3 \\ -x+y-2 z &=-11 \\ x+2 y+2 z &=-1 \end{aligned}\right. $$

If the voltage \(V\) in an electric circuit is held constant, the current \(I\) is inversely proportional to the resistance \(R\). If the current is 40 amperes when the resistance is 270 ohms, find the current when the resistance is 150 ohms.

Solve each system. $$ \left\\{\begin{array}{rr} 7 x+4 y & =10 \\ x-4 y+2 z & =6 \\ y-2 z & =-1 \end{array}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{r} x+y+z=9 \\ 3 x-y+z=-1 \\ -2 x+2 y-3 z=-2 \end{array}\right. $$

Because it is more efficient to produce larger numbers of items, the cost of producing a certain computer DVD is inversely proportional to the number produced. If 4000 can be produced at a cost of \(\$ 1.20\) each, find the cost per DVD when 6000 are produced.

Solve each system. $$ \left\\{\begin{array}{rr} x+2 y-z= & 5 \\ -3 x-2 y-3 z= & 11 \\ 4 x+4 y+5 z= & -18 \end{array}\right. $$

Evaluate each expression. $$ (-3)^{2} $$

The intensity \(I\) of light varies inversely as the square of the distance \(d\) from the light source. If the distance from the light source is doubled (see the figure), determine what happens to the intensity of light at the new location.

Evaluate each expression. $$ (-5)^{3} $$

Solve each system. $$ \left\\{\begin{aligned} 3 x-3 y+z &=-1 \\ 3 x-y-z &=3 \\ -6 x+4 y+3 z &=-8 \end{aligned}\right. $$

The maximum weight that a circular column can hold is inversely proportional to the square of its height. If an 8 -foot column can hold 2 tons, find how much weight a 10 -foot column can hold.

Evaluate each expression. $$ \left(\frac{2}{3}\right)^{2} $$

Solve each system. $$ \left\\{\begin{array}{l} \frac{3}{4} x-\frac{1}{3} y+\frac{1}{2} z=9 \\ \frac{1}{6} x+\frac{1}{3} y-\frac{1}{2} z=2 \\ \frac{1}{2} x-y+\frac{1}{2} z=2 \end{array}\right. $$

Write each statement as an equation. Use \(k\) as the constant of variation. \(x\) varies jointly as \(y\) and \(z\).

Evaluate each expression. $$ \left(\frac{3}{4}\right)^{3} $$

Solve each system. $$ \left\\{\begin{aligned} \frac{1}{3} x-\frac{1}{4} y+z &=-9 \\ \frac{1}{2} x-\frac{1}{3} y-\frac{1}{4} z &=-6 \\ x-\frac{1}{2} y-z &=-8 \end{aligned}\right. $$

Write each statement as an equation. Use \(k\) as the constant of variation. \(P\) varies jointly as \(R\) and the square of \(S\).

For Exercises 29 and 30 , the solutions have been started for you. The first few exercises are each modeled by a system of two linear equations in two variables. One number is two more than a second number. Twice the first is 4 less than 3 times the second. Find the numbers. 1\. UNDERSTAND the problem. Since we are looking for two numbers, let \(x=\) one number \(y=\) second number 2\. TRANSLATE. Since we have assigned two variables, we will translate the facts into two equations. (Fill in the blanks.) 3\. SOLVE the system and 4\. INTERPRET the results.

Perform each indicated operation. $$ \begin{aligned} &(-2)^{2}-(-3)+2(-1)\\\ &\text { 30. } 5^{2}-11+3(-5) \end{aligned} $$

Solve. See the Concept Check in this section. For the system \(\left\\{\begin{aligned} x \quad+z &=7 \\ y+2 z &=-6, \text { which is the correct corresponding matrix? } \\ 3 x-y \quad &=0 \end{aligned}\right.\) a. \(\left[\begin{array}{rrr}1 & 1 & 7 \\ 1 & 2 & -6 \\ 3 & -1 & 0\end{array}\right]\) b. \(\left[\begin{array}{rrrr}1 & 0 & 1 & 7 \\ 1 & 2 & 0 & -6 \\ 3 & -1 & 0 & 0\end{array}\right]\) C. \(\left[\begin{array}{rrrr}1 & 0 & 1 & 7 \\ 0 & 1 & 2 & -6 \\ 3 & -1 & 0 & 0\end{array}\right]\)

Write each statement as an equation. Use \(k\) as the constant of variation. \(r\) varies jointly as \(s\) and the cube of \(t\).