Chapter 9

Solve each system. $$ \left\\{\begin{aligned} x-y+z &=-4 \\ 3 x+2 y-z &=5 \\ -2 x+3 y-z &=15 \end{aligned}\right. $$

Graph the solutions of each system of linear inequalities. $$ \left\\{\begin{array}{l} y \geq x+1 \\ y \geq 3-x \end{array}\right. $$

Use matrices to solve each system of linear equations. See Example 1. $$ \left\\{\begin{array}{l} x+y=1 \\ x-2 y=4 \end{array}\right. $$

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=4\) when \(x=20\)

Solve each system. $$ \left\\{\begin{aligned} x+y-z &=-1 \\ -4 x-y+2 z &=-7 \\ 2 x-2 y-5 z &=7 \end{aligned}\right. $$

Graph the solutions of each system of linear inequalities..] $$ \left\\{\begin{array}{l} y \geq x-3 \\ y \geq-1-x \end{array}\right. $$

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=6\) when \(x=30\)

Graph the solutions of each system of linear inequalities.. $$ \left\\{\begin{array}{l} y<3 x-4 \\ y \leq x+2 \end{array}\right. $$

Use matrices to solve each system of linear equations. See Example 1. $$ \left\\{\begin{array}{l} x+3 y=2 \\ x+2 y=0 \end{array}\right. $$

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=6\) when \(x=4\)

Graph the solutions of each system of linear inequalities.. $$ \left\\{\begin{array}{l} y \leq 2 x+1 \\ y>x+2 \end{array}\right. $$

Solve each system. $$ \left\\{\begin{array}{ll} 5 x & =5 \\ 2 x+y & =4 \\ 3 x+y-4 z & =-15 \end{array}\right. $$

Use matrices to solve each system of linear equations. See Example 1. $$ \left\\{\begin{array}{l} 4 x-y=5 \\ 3 x-3 y=6 \end{array}\right. $$

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=12\) when \(x=8\)

Solve each system. $$ \left\\{\begin{array}{r} 2 x+2 y+z=1 \\ -x+y+2 z=3 \\ x+2 y+4 z=0 \end{array}\right. $$

Graph the solutions of each system of linear inequalities.. $$ \left\\{\begin{array}{l} y<-2 x-2 \\ y>x+4 \end{array}\right. $$

Use matrices to solve each system of linear equations. See Example 2. $$ \left\\{\begin{array}{r} x-2 y=4 \\ 2 x-4 y=4 \end{array}\right. $$

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=7\) when \(x=\frac{1}{2}\)

Solve each system. $$ \left\\{\begin{array}{r} 2 x-3 y+z=5 \\ x+y+z=0 \\ 4 x+2 y+4 z=4 \end{array}\right. $$

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

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=11\) when \(x=\frac{1}{3} \quad\)

Solve each system. $$ \left\\{\begin{array}{rr} x-2 y+z= & -5 \\ -3 x+6 y-3 z= & 15 \\ 2 x-4 y+2 z= & -10 \end{array}\right. $$

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

Use matrices to solve each system of linear equations. See Example 2. $$ \left\\{\begin{array}{l} 3 x-3 y=9 \\ 2 x-2 y=6 \end{array}\right. $$

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=0.2\) when \(x=0.8\)

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

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

Use matrices to solve each system of linear equations. See Example 2. $$ \left\\{\begin{array}{rr} 9 x-3 y= & 6 \\ -18 x+6 y= & -12 \end{array}\right. $$

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=0.4\) when \(x=2.5\)

Solve each system. $$ \left\\{\begin{aligned} 4 x-y+2 z &=5 \\ 2 y+z &=4 \\ 4 x+y+3 z &=10 \end{aligned}\right. $$

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

The weight of a synthetic ball varies directly with the cube of its radius. A ball with a radius of 2 inches weighs 1.20 pounds. Find the weight of a ball of the same material with a 3 -inch radius.

Solve each system. $$ \left\\{\begin{array}{r} 5 y-7 z=14 \\ 2 x+y+4 z=10 \\ 2 x+6 y-3 z=30 \end{array}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{aligned} -2 x &

At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above the water can see 7.4 miles, find how far a person 64 feet above the water can see. Round to the nearest tenth of a mile.

Solve each system. $$ \left\\{\begin{aligned} x &+5 z=0 \\ 5 x+y &=0 \\ y-3 z &=0 \end{aligned}\right. $$

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

Use matrices to solve each system of linear equations. See Example 3. $$ \left\\{\begin{aligned} 2 y-z &=-7 \\ x+4 y+z &=-4 \\ 5 x-y+2 z &=13 \end{aligned}\right. $$

The amount \(P\) of pollution varies directly with the population \(N\) of people. Kansas City has a population of 442,000 and produces 260,000 tons of pollutants. Find how many tons of pollution we should expect St. Louis to produce, if we know that its population is 348,000 . Round to the nearest whole ton. (Population Source: The World Almanac)

Solve each system. $$ \left\\{\begin{aligned} x-5 y &=0 \\ x &-z=0 \\ -x &+5 z=0 \end{aligned}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{array}{l} x \geq-3 \\ y \geq-2 \end{array}\right. $$

Use matrices to solve each system of linear equations. See Example 3. $$ \left\\{\begin{aligned} 4 y+3 z &=-2 \\ 5 x-4 y &=1 \\ -5 x+4 y+z &=-3 \end{aligned}\right. $$

Charles's law states that if the pressure \(P\) stays the same, the volume \(V\) of a gas is directly proportional to its temperature \(T\). If a balloon is filled with 20 cubic meters of a gas at a temperature of \(300 \mathrm{~K},\) find the new volume if the temperature rises \(360 \mathrm{~K}\) while the pressure stays the same.

Solve each system. $$ \left\\{\begin{array}{l} 6 x-5 z=17 \\ 5 x-y+3 z=-1 \\ 2 x+y \quad=-41 \end{array}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{array}{l} y \geq 1 \\ x<-3 \end{array}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=6\) when \(x=5\)

Solve each system. $$ \left\\{\begin{aligned} x+2 y &=6 \\ 7 x+3 y+z &=-33 \\ x-z &=16 \end{aligned}\right. $$

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{array}{l} y>2 \\ x \geq-1 \end{array}\right. $$

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{r} 3 y=6 \\ x+y=7 \end{array}\right. $$

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=20\) when \(x=9\)