Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} &\frac{3}{8} x-\frac{1}{2} y=\frac{7}{8}\\ &-6 x+8 y=-14 \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations.

$$\begin{array}{r} x-y=1 \\ -x+y=-1 \end{array}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} 2 x+y-z+3 w &=0 \\ 3 x-2 y+z-4 w &=-24 \\ x+y-z+w &=2 \\ x-y+2 z-5 w &=-16 \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} &\begin{array}{l} y=-2 x-2 z+1 \\ x=-2 y-z+2 \end{array}\\ &z=x-y \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} &2 x-y+3 z=0\\ &\begin{array}{l} x+2 y-z=5 \\ 2 y+z=1 \end{array} \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} &\frac{1}{2} x+\frac{3}{5} y=\frac{1}{4}\\ &10 x+12 y=5 \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} 3 x+y+3 z-1 &=0 \\ x+2 y-z-2 &=0 \\ 2 x-y+4 z-4 &=0 \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\ x-y-3 z-2 w &=2 \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} x+3 y-6 z &=7 \\ 2 x-y+z &=1 \\ x+2 y+2 z &=-1 \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{aligned} &4 x+2 y-3 z=6\\ &x-4 y+z=-4\\ &-x+2 z=2 \end{aligned}$$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, give the solution with y arbitrary. For systems in three variables with infinitely many solutions, give the solution with z arbitrary.

$$\begin{array}{c} 3 x+5 y-z+2=0 \\ 4 x-y+2 z-1=0 \\ -6 x-10 y+2 z=0 \end{array}$$

Rewrite $\frac{2(x+5)-4}{2}-x$ using the fewest terms possible. Justify each step with a property or operation.

A spacecraft is approaching a space station that is orbiting Earth. When the craft is 1000 km from the space station, reverse thrusters must be applied to begin braking. The time, t, in hours, required to reach a distance, d, in kilometres, from the space station while the thrusters are being fired can be modelled by $t=\log _{0.5}\left(\frac{d}{1000}\right)$. The docking sequence can be initialized once the craft is within 10 km of the station’s docking bay. a) How long after the reverse thrusters are first fired should docking procedures begin? b) What are the domain and range of this function? What do these features represent?

Determine the product of all values of k for which the polynomial equation $2 x^{3}-9 x^{2}+12 x-k=0$ has a double root.