9.8 Solving systems with cramer's rule  (Page 8/11)

 Page 8 / 11

$\left(10,-10,10\right)$

$\begin{array}{r}\hfill 5x+3y-z=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 3x-2y+4z=13\\ \hfill 4x+3y+5z=22\end{array}$

$\begin{array}{r}x+y+z=1\\ 2x+2y+2z=1\\ 3x+3y=2\end{array}$

No solutions exist.

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3x+2y-z=-10\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-y+2z=7\hfill \\ -x+3y+z=-2\hfill \end{array}$

$\left(-1,-2,3\right)$

$\begin{array}{r}\hfill 3x+4z=-11\\ \hfill x-2y=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 4y-z=-10\end{array}$

$\begin{array}{r}2x-3y+z=0\\ 2x+4y-3z=0\\ 6x-2y-z=0\end{array}$

$\left(x,\frac{8x}{5},\frac{14x}{5}\right)$

$\begin{array}{r}6x-4y-2z=2\\ 3x+2y-5z=4\\ 6y-7z=5\end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

11, 17, 33

A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at$15 for students, $12 for children, and$18 for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.

$\begin{array}{l}\begin{array}{l}\\ y={x}^{2}-7\end{array}\hfill \\ y=5x-13\hfill \end{array}$

$\left(2,-3\right),\left(3,2\right)$

$\begin{array}{l}\begin{array}{l}\\ y={x}^{2}-4\end{array}\hfill \\ y=5x+10\hfill \end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}=16\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=x-8\hfill \end{array}$

No solution

$\begin{array}{l}{x}^{2}+{y}^{2}=25\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y={x}^{2}+5\hfill \end{array}$

$\begin{array}{r}{x}^{2}+{y}^{2}=4\\ y-{x}^{2}=3\end{array}$

No solution

For the following exercises, graph the inequality.

$y>{x}^{2}-1$

$\frac{1}{4}{x}^{2}+{y}^{2}<4$

For the following exercises, graph the system of inequalities.

$\begin{array}{l}{x}^{2}+{y}^{2}+2x<3\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y>-{x}^{2}-3\hfill \end{array}$

$\begin{array}{l}{x}^{2}-2x+{y}^{2}-4x<4\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y<-x+4\hfill \end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}<1\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}^{2}

Partial Fractions

For the following exercises, decompose into partial fractions.

$\frac{-2x+6}{{x}^{2}+3x+2}$

$\frac{2}{x+2},\frac{-4}{x+1}$

$\frac{10x+2}{4{x}^{2}+4x+1}$

$\frac{7x+20}{{x}^{2}+10x+25}$

$\frac{7}{x+5},\frac{-15}{{\left(x+5\right)}^{2}}$

$\frac{x-18}{{x}^{2}-12x+36}$

$\frac{-{x}^{2}+36x+70}{{x}^{3}-125}$

$\frac{3}{x-5},\frac{-4x+1}{{x}^{2}+5x+25}$

$\frac{-5{x}^{2}+6x-2}{{x}^{3}+27}$

$\frac{{x}^{3}-4{x}^{2}+3x+11}{{\left({x}^{2}-2\right)}^{2}}$

$\frac{x-4}{\left({x}^{2}-2\right)},\frac{5x+3}{{\left({x}^{2}-2\right)}^{2}}$

$\frac{4{x}^{4}-2{x}^{3}+22{x}^{2}-6x+48}{x{\left({x}^{2}+4\right)}^{2}}$

Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

$A=\left[\begin{array}{rr}\hfill 4& \hfill -2\\ \hfill 1& \hfill 3\end{array}\right],B=\left[\begin{array}{rrr}\hfill 6& \hfill 7& \hfill -3\\ \hfill 11& \hfill -2& \hfill 4\end{array}\right],C=\left[\begin{array}{r}\hfill \begin{array}{cc}6& 7\\ 11& -2\end{array}\\ \hfill \begin{array}{cc}14& 0\end{array}\end{array}\right],D=\left[\begin{array}{rrr}\hfill 1& \hfill -4& \hfill 9\\ \hfill 10& \hfill 5& \hfill -7\\ \hfill 2& \hfill 8& \hfill 5\end{array}\right],E=\left[\begin{array}{rrr}\hfill 7& \hfill -14& \hfill 3\\ \hfill 2& \hfill -1& \hfill 3\\ \hfill 0& \hfill 1& \hfill 9\end{array}\right]$

$-4A$

$\left[\begin{array}{cc}-16& 8\\ -4& -12\end{array}\right]$

$10D-6E$

$B+C$

undefined; dimensions do not match

$AB$

$BA$

undefined; inner dimensions do not match

$BC$

$CB$

$\left[\begin{array}{ccc}113& 28& 10\\ 44& 81& -41\\ 84& 98& -42\end{array}\right]$

$DE$

$ED$

$\left[\begin{array}{ccc}-127& -74& 176\\ -2& 11& 40\\ 28& 77& 38\end{array}\right]$

$EC$

$CE$

undefined; inner dimensions do not match

${A}^{3}$

Solving Systems with Gaussian Elimination

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

$\begin{array}{l}x-3z=7\\ y+2z=-5\text{\hspace{0.17em}}\end{array}$ with infinite solutions

For the following exercises, write the augmented matrix from the system of linear equations.

$\begin{array}{l}\\ \begin{array}{r}\hfill -2x+2y+z=7\\ \hfill 2x-8y+5z=0\\ \hfill 19x-10y+22z=3\end{array}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4x+2y-3z=14\hfill \\ -12x+3y+z=100\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9x-6y+2z=31\hfill \end{array}$

$\begin{array}{r}\hfill x+3z=12\text{\hspace{0.17em}}\\ \hfill -x+4y=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill y+2z=-7\end{array}$

For the following exercises, solve the system of linear equations using Gaussian elimination.

$\begin{array}{r}3x-4y=-7\\ -6x+8y=14\end{array}$

$\begin{array}{r}3x-4y=1\\ -6x+8y=6\end{array}$

No solutions exist.

$\begin{array}{l}\begin{array}{l}\\ -1.1x-2.3y=6.2\end{array}\hfill \\ -5.2x-4.1y=4.3\hfill \end{array}$

$\begin{array}{r}\hfill 2x+3y+2z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -4x-6y-4z=-2\\ \hfill 10x+15y+10z=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

No solutions exist.

$\begin{array}{r}\hfill -x+2y-4z=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 3y+8z=-4\\ \hfill -7x+y+2z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.

$\left[\begin{array}{rr}\hfill -0.2& \hfill 1.4\\ \hfill 1.2& \hfill -0.4\end{array}\right]$

$\frac{1}{8}\left[\begin{array}{cc}2& 7\\ 6& 1\end{array}\right]$

$\left[\begin{array}{rr}\hfill \frac{1}{2}& \hfill -\frac{1}{2}\\ \hfill -\frac{1}{4}& \hfill \frac{3}{4}\end{array}\right]$

$\left[\begin{array}{ccc}12& 9& -6\\ -1& 3& 2\\ -4& -3& 2\end{array}\right]$

No inverse exists.

$\left[\begin{array}{ccc}2& 1& 3\\ 1& 2& 3\\ 3& 2& 1\end{array}\right]$

preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4
hil
hi
A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center