1-4 Exercises Guided Practice Geometry

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11.one Systems of Linear Equations: Two Variables

2.

The solution to the organisation is the ordered pair $\left(\mathrm{-5},3\right).$

6.

No solution. It is an inconsistent organisation.

seven.

The organization is dependent and then there are infinite solutions of the course $(x,2\mathrm{10}+5).$

viii.

700 children, 950 adults

11.two Systems of Linear Equations: Three Variables

3.

Space number of solutions of the form $\left(\mathrm{ten},4\mathrm{ten}\mathrm{-xi},\mathrm{-5}\mathrm{10}+18\right).$

11.3 Systems of Nonlinear Equations and Inequalities: 2 Variables

1.

$\left(-\frac{\mathrm{one}}{2},\frac{1}{\mathrm{ii}}\right)$ and $\left(\mathrm{two},\mathrm{viii}\right)$

3.

$\left\{\left(1,\mathrm{three}\right),\left(\mathrm{one},\mathrm{-3}\right),\left(\mathrm{-1},3\right),\left(\mathrm{-ane},\mathrm{-three}\right)\right\}$

11.4 Partial Fractions

1.

$\frac{\mathrm{three}}{\mathrm{ten}\mathrm{-iii}}-\frac{\mathrm{ii}}{\mathrm{ten}\mathrm{-2}}$

2.

$\frac{6}{\mathrm{ten}\mathrm{-1}}-\frac{\mathrm{five}}{{\left(\mathrm{10}\mathrm{-1}\right)}^2}$

3.

$\frac{\mathrm{iii}}{x\mathrm{-ane}}+\frac{2x\mathrm{-4}}{x^{\mathrm{ii}}+1}$

4.

$\frac{\mathrm{ten}\mathrm{-2}}{{\mathrm{ten}}^2\mathrm{-two}\mathrm{ten}+\mathrm{three}}+\frac{2x+1}{{\left(x^2\mathrm{-2}x+\mathrm{iii}\right)}^2}$

11.5 Matrices and Matrix Operations

1.

$A+B=\left[\begin{array}{c}2\\ 1\\ \mathrm{i}\end{array}\begin{array}{c}\mathrm{half-dozen}\\ \text{}\text{}\text{}0\\ \mathrm{-3}\end{array}\right]+\left[\begin{array}{c}3\\ 1\\ \mathrm{-4}\end{array}\begin{array}{c}\mathrm{-2}\\ 5\\ \mathrm{three}\end{array}\right]$
$=\left[\begin{array}{c}2+3\\ 1+\mathrm{ane}\\ \mathrm{i}+(\mathrm{-iv})\end{array}\begin{array}{c}6+(\mathrm{-2})\\ 0+5\\ \mathrm{-three}+3\end{array}\right]=\left[\begin{array}{c}\mathrm{five}\\ \mathrm{two}\\ \mathrm{-three}\end{array}\begin{array}{c}4\\ 5\\ 0\end{array}\right]$

2.

$\mathrm{-ii}B=\left[\begin{array}{cc}\mathrm{-8}& \mathrm{-two}\\ \mathrm{-6}& \mathrm{-4}\end{array}\right]$

eleven.6 Solving Systems with Gaussian Elimination

1.

$\left[\begin{array}{cc}4& \mathrm{-3}\\ 3& 2\end{array}|\begin{array}{c}11\\ \mathrm{four}\end{array}\right]$

two.

$\begin{array}{c}x-y+z=5\\ 2x-y+3z=1\\ y+z=\mathrm{-9}\end{array}$

4.

$\left[\begin{array}{ccc}1& -\frac{5}{2}& \frac{5}{\mathrm{ii}}\\ \text{}0& \mathrm{ane}& \mathrm{v}\\ 0& 0& 1\end{array}|\begin{array}{c}\frac{17}{\mathrm{ii}}\\ 9\\ \mathrm{ii}\end{array}\right]$

six.

$150,000 at 7%, $750,000 at eight%, $600,000 at 10%

11.vii Solving Systems with Inverses

one.

$$\begin{array}{l}AB=\left[\begin{array}{rrr}\hfill 1& \hfill & \hfill \mathrm{four}\\ \hfill \mathrm{-1}& \hfill & \hfill \mathrm{-3}\end{array}\right]\begin{array}{r}\hfill \end{array}\left[\begin{array}{rrr}\hfill \mathrm{-three}& \hfill & \hfill \mathrm{-iv}\\ \hfill 1& \hfill & \hfill 1\end{array}\right]=\left[\begin{array}{rrr}\hfill 1(\mathrm{-3})+4(1)& \hfill & \hfill 1(\mathrm{-4})+4(1)\\ \hfill \mathrm{-1}(\mathrm{-3})+\mathrm{-3}(1)& \hfill & \hfill \mathrm{-1}(\mathrm{-iv})+\mathrm{-3}(1)\end{array}\right]=\left[\begin{array}{rrr}\hfill 1& \hfill & \hfill 0\\ \hfill 0& \hfill & \hfill 1\end{array}\right]\hfill \\ BA=\left[\begin{array}{rrr}\hfill \mathrm{-3}& \hfill & \hfill \mathrm{-4}\\ \hfill 1& \hfill & \hfill \mathrm{one}\end{array}\right]\begin{array}{r}\hfill \end{array}\left[\begin{array}{rrr}\hfill 1& \hfill & \hfill 4\\ \hfill \mathrm{-one}& \hfill & \hfill \mathrm{-iii}\end{array}\right]=\left[\begin{array}{rrr}\hfill \mathrm{-3}(\mathrm{i})+\mathrm{-4}(\mathrm{-1})& \hfill & \hfill \mathrm{-3}(\mathrm{four})+\mathrm{-4}(\mathrm{-3})\\ \hfill 1(1)+1(\mathrm{-1})& \hfill & \hfill \mathrm{one}(4)+\mathrm{ane}(\mathrm{-three})\end{array}\right]=\left[\begin{array}{rrr}\hfill 1& \hfill & \hfill 0\\ \hfill 0& \hfill & \hfill 1\end{array}\right]\hfill \end{array}$$

ii.

$A^{\mathrm{-i}}=\left[\begin{array}{cc}\frac{3}{5}& \frac{\mathrm{i}}{5}\\ -\frac{\mathrm{ii}}{\mathrm{five}}& \frac{1}{\mathrm{five}}\end{array}\right]$

three.

$A^{\mathrm{-i}}=\left[\begin{array}{ccc}1& 1& \mathrm{ii}\\ 2& 4& \mathrm{-iii}\\ \mathrm{three}& \mathrm{vi}& \mathrm{-v}\end{array}\right]$

4.

$\mathrm{Ten}=\left[\begin{array}{c}\mathrm{four}\\ 38\\ 58\end{array}\right]$

11.8 Solving Systems with Cramer'due south Dominion

3.

$\left(-2,\frac{\mathrm{three}}{5},\frac{12}{\mathrm{v}}\right)$

xi.1 Section Exercises

1.

No, you lot can either have zero, i, or infinitely many. Examine graphs.

3.

This ways there is no realistic interruption-even indicate. By the fourth dimension the visitor produces i unit of measurement they are already making profit.

five.

You tin can solve by substitution (isolating $x$ or $y$ ), graphically, or by add-on.

15.

$\left(-\frac{3}{\mathrm{v}},0\right)$

19.

$\left(\frac{72}{5},\frac{132}{\mathrm{v}}\right)$

23.

$\left(-\frac{1}{2},\frac{\mathrm{ane}}{10}\right)$

27.

$\left(-\frac{1}{\mathrm{five}},\frac{2}{\mathrm{three}}\right)$

29.

$\left(x,\frac{x+3}{\mathrm{two}}\right)$

33.

$\left(\frac{1}{2},\frac{1}{8}\right)$

35.

$\left(\frac{1}{6},0\right)$

37.

$\left(x,\mathrm{ii}(7x\mathrm{-half-dozen})\right)$

39.

$\left(-\frac{5}{\mathrm{half\; dozen}},\frac{\mathrm{iv}}{3}\right)$

41.

Consistent with one solution

43.

Consequent with 1 solution

45.

Dependent with infinitely many solutions

47.

$\left(\mathrm{-3.08},4.91\right)$

49.

$\left(\mathrm{-1.52},2.29\right)$

51.

$\left(\frac{A+B}{2},\frac{A-B}{2}\right)$

53.

$\left(\frac{\mathrm{-i}}{A-B},\frac{A}{A-B}\right)$

55.

$\left(\frac{CE-BF}{BD-AE},\frac{AF-CD}{BD-A\mathrm{Due\; east}}\right)$

57.

They never profit.

59.

$(\mathrm{ane},250,100,000)$

61.

The numbers are 7.five and 20.5.

65.

790 sophomores, 805 freshman

69.

10 gallons of x% solution, 15 gallons of 60% solution

71.

Swan Superlative: $750,000, Riverside: $350,000

73.

$12,500 in the outset business relationship, $10,500 in the second account.

75.

High-tops: 45, Depression-tops: xv

77.

Infinitely many solutions. We need more than data.

eleven.2 Section Exercises

ane.

No, there tin be only i, zilch, or infinitely many solutions.

3.

Non necessarily. At that place could be zero, one, or infinitely many solutions. For example, $\left(0,0,0\right)$ is not a solution to the system below, merely that does not mean that it has no solution.

$\begin{array}{l}2\mathrm{10}+3y\mathrm{-6}z=1\hfill \\ \mathrm{-four}\mathrm{ten}\mathrm{-6}y+12z=\mathrm{-two}\hfill \\ x+2y+\mathrm{v}z=10\hfill \end{array}$

v.

Every organization of equations can be solved graphically, past substitution, and past addition. Withal, systems of three equations become very complex to solve graphically so other methods are usually preferable.

11.

$\left(\mathrm{-1},4,2\right)$

13.

$\left(-\frac{85}{107},\frac{312}{107},\frac{191}{107}\right)$

15.

$\left(1,\frac{1}{2},0\right)$

17.

$\left(\mathrm{four},\mathrm{-6},\mathrm{ane}\right)$

nineteen.

$\left(x,\frac{\mathrm{ane}}{27}(65\mathrm{-16}x),\frac{\mathrm{ten}+28}{27}\right)$

21.

$\left(-\frac{45}{\mathrm{thirteen}},\frac{17}{13},\mathrm{-2}\right)$

27.

$\left(\frac{4}{\mathrm{vii}},-\frac{1}{7},-\frac{3}{\mathrm{seven}}\right)$

29.

$\left(7,\mathrm{twenty},16\right)$

31.

$\left(\mathrm{-six},2,1\right)$

33.

$\left(5,12,\mathrm{xv}\right)$

35.

$\left(\mathrm{-5},\mathrm{-5},\mathrm{-five}\right)$

37.

$\left(10,10,\mathrm{x}\right)$

39.

$\left(\frac{\mathrm{ane}}{2},\frac{\mathrm{ane}}{5},\frac{4}{5}\right)$

41.

$\left(\frac{\mathrm{ane}}{\mathrm{two}},\frac{2}{5},\frac{4}{\mathrm{v}}\right)$

47.

$\left(\frac{128}{557},\frac{23}{557},\frac{28}{557}\right)$

49.

$\left(6,\mathrm{-i},0\right)$

53.

70 grandparents, 140 parents, 190 children

55.

Your share was $19.95, Sarah'south share was $40, and your other roommate's share was $22.05.

57.

There are infinitely many solutions; we need more information

59.

500 students, 225 children, and 450 adults

61.

The BMW was $49,636, the Jeep was $42,636, and the Toyota was $47,727.

63.

$400,000 in the account that pays 3% interest, $500,000 in the account that pays four% interest, and $100,000 in the account that pays 2% involvement.

65.

The Usa consumed 26.3%, Japan 7.i%, and China 6.4% of the world's oil.

67.

Kingdom of saudi arabia imported 16.8%, Canada imported 15.ane%, and Mexico 15.0%

69.

Birds were nineteen.3%, fish were xviii.six%, and mammals were 17.1% of endangered species

11.3 Section Exercises

ane.

A nonlinear system could be representative of ii circles that overlap and intersect in two locations, hence 2 solutions. A nonlinear system could exist representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions.

3.

No. There does not need to be a feasible region. Consider a system that is bounded by ii parallel lines. One inequality represents the region in a higher place the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.

5.

Choose any number between each solution and plug into $C(x)$ and $R(\mathrm{ten}).$ If $C(\mathrm{10})<R(\mathrm{10}),$ then there is profit.

vii.

$\left(0,\mathrm{-3}\right),\left(\mathrm{iii},0\right)$

nine.

$\left(-\frac{3\sqrt{2}}{\mathrm{ii}},\frac{\mathrm{three}\sqrt{2}}{2}\right),\left(\frac{3\sqrt{2}}{2},-\frac{3\sqrt{2}}{2}\right)$

11.

$\left(\mathrm{-3},0\right),\left(3,0\right)$

thirteen.

$\left(\frac{1}{4},-\frac{\sqrt{62}}{8}\right),\left(\frac{1}{\mathrm{iv}},\frac{\sqrt{62}}{8}\right)$

15.

$\left(-\frac{\sqrt{398}}{4},\frac{199}{4}\right),\left(\frac{\sqrt{398}}{4},\frac{199}{\mathrm{iv}}\right)$

17.

$\left(0,2\right),\left(\mathrm{one},3\right)$

xix.

$\left(-\sqrt{\frac{1}{2}\left(\sqrt{5}\mathrm{-one}\right)},\frac{1}{\mathrm{two}}\left(\mathrm{one}-\sqrt{5}\right)\right),\left(\sqrt{\frac{\mathrm{i}}{\mathrm{two}}\left(\sqrt{5}\mathrm{-1}\right)},\frac{1}{2}\left(1-\sqrt{5}\right)\right)$

31.

$\left(-\frac{\sqrt{2}}{\mathrm{two}},-\frac{\sqrt{\mathrm{ii}}}{\mathrm{ii}}\right),\left(-\frac{\sqrt{\mathrm{two}}}{2},\frac{\sqrt{2}}{\mathrm{two}}\right),\left(\frac{\sqrt{2}}{\mathrm{ii}},-\frac{\sqrt{2}}{2}\right),\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{\mathrm{two}}\right)$

35.

$\left(-\sqrt{7},\mathrm{-3}\right),\left(-\sqrt{7},3\right),\left(\sqrt{7},\mathrm{-3}\right),\left(\sqrt{\mathrm{seven}},3\right)$

37.

$\left(-\sqrt{\frac{\mathrm{i}}{2}\left(\sqrt{73}\mathrm{-5}\right)},\frac{\mathrm{ane}}{\mathrm{ii}}\left(7-\sqrt{73}\right)\right),\left(\sqrt{\frac{1}{2}\left(\sqrt{73}\mathrm{-5}\right)},\frac{1}{\mathrm{ii}}\left(\mathrm{vii}-\sqrt{73}\right)\right)$

49.

$\left(\mathrm{-ii}\sqrt{\frac{70}{383}},\mathrm{-2}\sqrt{\frac{35}{29}}\right),\left(\mathrm{-2}\sqrt{\frac{\mathrm{seventy}}{383}},2\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{70}{383}},\mathrm{-2}\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{\mathrm{lxx}}{383}},2\sqrt{\frac{35}{29}}\right)$

53.

$\mathrm{10}=0,y>0$ and $0<x<1,\sqrt{x}<y<\frac{\mathrm{ane}}{x}$

xi.4 Section Exercises

1.

No, a caliber of polynomials can but be decomposed if the denominator can be factored. For example, $\frac{1}{{\mathrm{10}}^{\mathrm{ii}}+1}$ cannot be decomposed because the denominator cannot exist factored.

three.

Graph both sides and ensure they are equal.

v.

If we choose $x=\mathrm{-1},$ so the B-term disappears, letting us immediately know that $A=3.$ We could alternatively plug in $x=-\frac{5}{\mathrm{three}}$ , giving the states a B-value of $\mathrm{-2.}$

7.

$\frac{8}{x+3}-\frac{\mathrm{v}}{\mathrm{10}\mathrm{-8}}$

9.

$\frac{1}{x+5}+\frac{9}{x+2}$

11.

$\frac{3}{5x\mathrm{-2}}+\frac{4}{\mathrm{iv}x\mathrm{-i}}$

thirteen.

$\frac{5}{2\left(\mathrm{ten}+\mathrm{iii}\right)}+\frac{5}{\mathrm{two}\left(\mathrm{10}\mathrm{-3}\right)}$

15.

$\frac{3}{x+2}+\frac{3}{\mathrm{10}\mathrm{-two}}$

17.

$\frac{\mathrm{nine}}{5\left(x+2\right)}+\frac{11}{5\left(\mathrm{ten}\mathrm{-3}\right)}$

19.

$\frac{\mathrm{viii}}{x\mathrm{-three}}-\frac{\mathrm{v}}{x\mathrm{-ii}}$

21.

$\frac{1}{x\mathrm{-2}}+\frac{2}{{\left(x\mathrm{-2}\right)}^2}$

23.

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

25.

$-\frac{1}{\mathrm{10}\mathrm{-7}}-\frac{2}{{\left(x\mathrm{-seven}\right)}^2}$

27.

$\frac{4}{\mathrm{10}}-\frac{3}{2\left(\mathrm{10}+1\right)}+\frac{\mathrm{seven}}{2{\left(x+1\right)}^2}$

29.

$\frac{4}{\mathrm{10}}+\frac{2}{{\mathrm{10}}^2}-\frac{\mathrm{three}}{\mathrm{three}\mathrm{10}+\mathrm{ii}}+\frac{7}{2{\left(3\mathrm{10}+2\right)}^{\mathrm{ii}}}$

31.

$\frac{x+1}{x^{\mathrm{two}}+\mathrm{ten}+\mathrm{three}}+\frac{3}{x+\mathrm{ii}}$

33.

$\frac{4\mathrm{-3}x}{x^2+3x+\mathrm{viii}}+\frac{1}{\mathrm{ten}\mathrm{-1}}$

35.

$\frac{\mathrm{two}\mathrm{10}\mathrm{-1}}{x^2+\mathrm{half\; dozen}x+1}+\frac{2}{x+\mathrm{iii}}$

37.

$\frac{\mathrm{i}}{x^2+\mathrm{ten}+1}+\frac{4}{x\mathrm{-ane}}$

39.

$\frac{2}{x^2\mathrm{-3}\mathrm{ten}+9}+\frac{3}{x+3}$

41.

$-\frac{1}{4x^{\mathrm{ii}}+6x+9}+\frac{1}{2x\mathrm{-3}}$

43.

$\frac{1}{x}+\frac{1}{x+\mathrm{vi}}-\frac{\mathrm{iv}x}{x^2\mathrm{-6}x+36}$

45.

$\frac{\mathrm{10}+\mathrm{vi}}{x^2+1}+\frac{\mathrm{iv}x+3}{{\left(x^2+1\right)}^2}$

47.

$\frac{x+\mathrm{one}}{\mathrm{ten}+2}+\frac{\mathrm{two}\mathrm{ten}+3}{{\left(x+2\right)}^2}$

49.

$\frac{1}{x^{\mathrm{two}}+3x+25}-\frac{3x}{{\left(x^2+3x+25\right)}^{\mathrm{ii}}}$

51.

$\frac{1}{\mathrm{viii}\mathrm{10}}-\frac{\mathrm{ten}}{8\left(x^{\mathrm{two}}+4\right)}+\frac{10-x}{2{\left(x^2+4\right)}^{\mathrm{two}}}$

53.

$-\frac{16}{\mathrm{ten}}-\frac{\mathrm{ix}}{x^{\mathrm{ii}}}+\frac{16}{x\mathrm{-1}}-\frac{\mathrm{seven}}{{\left(\mathrm{10}\mathrm{-i}\right)}^2}$

55.

$\frac{1}{x+\mathrm{i}}-\frac{2}{{\left(x+1\right)}^2}+\frac{\mathrm{five}}{{\left(x+1\right)}^{\mathrm{iii}}}$

57.

$\frac{5}{\mathrm{ten}\mathrm{-ii}}-\frac{\mathrm{iii}}{10\left(x+2\right)}+\frac{7}{\mathrm{ten}+\mathrm{eight}}-\frac{\mathrm{vii}}{10\left(\mathrm{ten}\mathrm{-8}\right)}$

59.

$-\frac{5}{\mathrm{iv}x}-\frac{5}{\mathrm{two}\left(\mathrm{10}+2\right)}+\frac{11}{\mathrm{two}\left(\mathrm{10}+4\right)}+\frac{5}{4\left(x+4\right)}$

11.5 Section Exercises

1.

No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add together the post-obit two matrices because the first is a $2\times \mathrm{two}$ matrix and the second is a $2\times \mathrm{iii}$ matrix. $\left[\begin{array}{cc}\mathrm{i}& 2\\ \mathrm{three}& \mathrm{iv}\end{array}\right]+\left[\begin{array}{ccc}\mathrm{half\; dozen}& 5& \mathrm{iv}\\ 3& \mathrm{two}& \mathrm{one}\end{array}\right]$ has no sum.

3.

Yes, if the dimensions of $A$ are $\mathrm{thousand}\times n$ and the dimensions of $B$ are $\mathrm{northward}\times m,$ both products will exist divers.

five.

Non necessarily. To find $AB,$ we multiply the first row of $A$ past the first cavalcade of $B$ to get the get-go entry of $AB.$ To observe $BA,$ we multiply the first row of $B$ past the showtime column of $A$ to get the first entry of $BA.$ Thus, if those are unequal, and then the matrix multiplication does not commute.

7.

$\left[\begin{array}{cc}\mathrm{xi}& 19\\ 15& 94\\ 17& 67\end{array}\right]$

9.

$\left[\begin{array}{cc}\mathrm{-4}& \mathrm{two}\\ 8& 1\end{array}\right]$

11.

Undidentified; dimensions practise non match

13.

$\left[\begin{array}{cc}\mathrm{ix}& 27\\ 63& 36\\ 0& 192\end{array}\right]$

15.

$\left[\begin{array}{cccc}\mathrm{-64}& \mathrm{-12}& \mathrm{-28}& \mathrm{-72}\\ \mathrm{-360}& \mathrm{-20}& \mathrm{-12}& \mathrm{-116}\end{array}\right]$

17.

$\left[\begin{array}{ccc}\mathrm{i},800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& \mathrm{two},100\end{array}\right]$

19.

$\left[\begin{array}{cc}\mathrm{twenty}& 102\\ 28& 28\end{array}\right]$

21.

$\left[\begin{array}{ccc}60& 41& 2\\ \mathrm{-16}& 120& \mathrm{-216}\end{array}\right]$

23.

$\left[\begin{array}{ccc}\mathrm{-68}& 24& 136\\ \mathrm{-54}& \mathrm{-12}& 64\\ \mathrm{-57}& 30& 128\end{array}\right]$

25.

Undefined; dimensions do not match.

27.

$\left[\begin{array}{ccc}\mathrm{-viii}& 41& \mathrm{-iii}\\ \mathrm{twoscore}& \mathrm{-fifteen}& \mathrm{-14}\\ 4& 27& 42\end{array}\right]$

29.

$\left[\begin{array}{ccc}\mathrm{-840}& 650& \mathrm{-530}\\ 330& 360& 250\\ \mathrm{-10}& 900& 110\end{array}\right]$

31.

$\left[\begin{array}{cc}\mathrm{-350}& \mathrm{one},050\\ 350& 350\end{array}\right]$

33.

Undefined; inner dimensions practice not match.

35.

$\left[\begin{array}{cc}\mathrm{i},400& 700\\ \mathrm{-1},400& 700\end{array}\right]$

37.

$\left[\begin{array}{cc}332,500& 927,500\\ \mathrm{-227},500& 87,500\end{array}\right]$

39.

$\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right]$

41.

$\left[\begin{array}{ccc}\mathrm{-2}& 3& 4\\ \mathrm{-seven}& 9& \mathrm{-7}\end{array}\right]$

43.

$\left[\begin{array}{ccc}\mathrm{-4}& 29& 21\\ \mathrm{-27}& \mathrm{-iii}& \mathrm{ane}\end{array}\right]$

45.

$\left[\begin{array}{ccc}\mathrm{-3}& \mathrm{-2}& \mathrm{-2}\\ \mathrm{-28}& 59& 46\\ \mathrm{-four}& 16& 7\end{array}\right]$

47.

$\left[\begin{array}{ccc}1& \mathrm{-18}& \mathrm{-9}\\ \mathrm{-198}& 505& 369\\ \mathrm{-72}& 126& 91\end{array}\right]$

49.

$\left[\begin{array}{cc}0& 1.6\\ \mathrm{nine}& \mathrm{-i}\end{array}\right]$

51.

$\left[\begin{array}{ccc}2& 24& \mathrm{-4.5}\\ 12& 32& \mathrm{-9}\\ \mathrm{-eight}& 64& 61\end{array}\right]$

53.

$\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& \mathrm{ten}\end{array}\right]$

55.

$\left[\begin{array}{ccc}\mathrm{i}& 0& 0\\ 0& 1& 0\\ 0& 0& \mathrm{i}\end{array}\right]$

57.

$\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{i}& 0\\ 0& 0& 1\end{array}\right]$

59.

$B^n=\{\begin{array}{l}\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{i}& 0\\ 0& 0& 1\end{array}\right],\mathrm{due\; north}\text{even,}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& \mathrm{ane}& 0\end{array}\right],n\text{odd}\text{.}\end{array}$

11.vi Section Exercises

1.

Yes. For each row, the coefficients of the variables are written beyond the corresponding row, and a vertical bar is placed; then the constants are placed to the correct of the vertical bar.

3.

No, there are numerous correct methods of using row operations on a matrix. Ii possible means are the following: (one) Interchange rows i and 2. Then $R_2=R_2\mathrm{-nine}{R}_1.$ (2) $R_2=R_{\mathrm{one}}\mathrm{-9}{R}_2.$ Then divide row 1 by 9.

5.

No. A matrix with 0 entries for an entire row would accept either nil or infinitely many solutions.

vii.

$\left[\begin{array}{rrrr}\hfill 0& \hfill & \hfill \mathrm{sixteen}& \hfill \\ \hfill 9& \hfill & \hfill \mathrm{-1}& \hfill \end{array}|\begin{array}{rr}\hfill & \hfill 4\\ \hfill & \hfill 2\end{array}\right]$

nine.

$\left[\begin{array}{rrrrrr}\hfill 1& \hfill & \hfill 5& \hfill & \hfill 8& \hfill \\ \hfill 12& \hfill & \hfill 3& \hfill & \hfill 0& \hfill \\ \hfill 3& \hfill & \hfill 4& \hfill & \hfill \mathrm{ix}& \hfill \end{array}|\begin{array}{rr}\hfill & \hfill \mathrm{sixteen}\\ \hfill & \hfill 4\\ \hfill & \hfill \mathrm{-vii}\end{array}\right]$

11.

$\begin{array}{l}\mathrm{-2}\mathrm{ten}+5y=5\\ 6x\mathrm{-eighteen}y=26\end{array}$

13.

$\begin{array}{l}3\mathrm{10}+\mathrm{two}y=3\\ -x\mathrm{-nine}y+4z=\mathrm{-1}\\ 8\mathrm{10}+5y+7z=\mathrm{eight}\end{array}$

15.

$\begin{array}{l}4\mathrm{10}+5y\mathrm{-two}z=12\hfill \\ y+58z=\mathrm{ii}\hfill \\ 8x+7y\mathrm{-3}z=\mathrm{-v}\hfill \end{array}$

25.

$\left(\frac{\mathrm{i}}{5},\frac{1}{2}\right)$

27.

$\left(\mathrm{ten},\frac{4}{15}(\mathrm{five}\mathrm{10}+\mathrm{ane})\right)$

31.

$\left(\frac{196}{39},-\frac{5}{13}\right)$

33.

$\left(31,\mathrm{-42},87\right)$

35.

$\left(\frac{21}{40},\frac{1}{20},\frac{9}{\mathrm{viii}}\right)$

37.

$\left(\frac{\mathrm{eighteen}}{13},\frac{15}{13},-\frac{15}{13}\right)$

39.

$\left(x,y,\frac{1}{2}(\mathrm{i}\mathrm{-2}\mathrm{ten}\mathrm{-3}y)\right)$

41.

$\left(x,-\frac{\mathrm{10}}{2},\mathrm{-i}\right)$

43.

$\left(125,\mathrm{-25},0\right)$

45.

$\left(8,\mathrm{one},\mathrm{-2}\right)$

49.

$\left(x,\frac{31}{28}-\frac{\mathrm{iii}x}{\mathrm{four}},\frac{1}{28}(\mathrm{-7}x\mathrm{-3})\right)$

53.

860 cerise velvet, 1,340 chocolate

55.

4% for business relationship ane, six% for business relationship 2

59.

Banana was 3%, pumpkin was seven%, and rocky route was 2%

61.

100 almonds, 200 cashews, 600 pistachios

11.7 Department Exercises

1.

If $A^{\mathrm{-1}}$ is the inverse of $A,$ then $A{A}^{\mathrm{-ane}}=I,$ the identity matrix. Since $A$ is besides the inverse of $A^{\mathrm{-i}},A^{\mathrm{-one}}A=I.$ You tin also check by proving this for a $\mathrm{two}\times \mathrm{two}$ matrix.

3.

No, because $ad$ and $bc$ are both 0, so $ad-bc=0,$ which requires us to separate by 0 in the formula.

5.

Yes. Consider the matrix $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$ The inverse is institute with the following calculation: $A^{\mathrm{-i}}=\frac{1}{0(0)\mathrm{-ane}(1)}\left[\begin{array}{cc}0& \mathrm{-1}\\ \mathrm{-1}& 0\end{array}\right]=\left[\begin{array}{cc}0& \mathrm{one}\\ \mathrm{one}& 0\end{array}\right].$

7.

$AB=BA=\left[\begin{array}{cc}1& 0\\ 0& \mathrm{one}\end{array}\right]=I$

nine.

$AB=BA=\left[\begin{array}{cc}\mathrm{one}& 0\\ 0& \mathrm{ane}\end{array}\right]=I$

11.

$AB=BA=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{i}& 0\\ 0& 0& 1\end{array}\right]=I$

13.

$\frac{1}{29}\left[\begin{array}{cc}9& \mathrm{ii}\\ \mathrm{-1}& 3\end{array}\right]$

15.

$\frac{1}{69}\left[\begin{array}{cc}\mathrm{-2}& 7\\ \mathrm{ix}& 3\end{array}\right]$

xix.

$\frac{4}{7}\left[\begin{array}{cc}\mathrm{0.five}& 1.5\\ \mathrm{i}& \mathrm{-0.five}\end{array}\right]$

21.

$\frac{1}{17}\left[\begin{array}{ccc}\mathrm{-5}& 5& \mathrm{-3}\\ 20& \mathrm{-iii}& 12\\ \mathrm{ane}& \mathrm{-ane}& \mathrm{four}\end{array}\right]$

23.

$\frac{1}{209}\left[\begin{array}{ccc}47& \mathrm{-57}& 69\\ 10& 19& \mathrm{-12}\\ \mathrm{-24}& 38& \mathrm{-13}\end{array}\right]$

25.

$\left[\begin{array}{ccc}\mathrm{eighteen}& \mathrm{threescore}& \mathrm{-168}\\ \mathrm{-56}& \mathrm{-140}& 448\\ 40& 80& \mathrm{-280}\end{array}\right]$

31.

$\left(\frac{1}{\mathrm{iii}},-\frac{5}{2}\right)$

33.

$\left(-\frac{\mathrm{ii}}{3},-\frac{11}{6}\right)$

35.

$\left(\mathrm{seven},\frac{1}{2},\frac{1}{\mathrm{five}}\right)$

37.

$\left(\mathrm{five},0,\mathrm{-ane}\right)$

39.

$\frac{1}{34}\left(\mathrm{-35},\mathrm{-97},\mathrm{-154}\right)$

41.

$\frac{1}{690}\left(65,\mathrm{-1136},\mathrm{-229}\right)$

43.

$\left(-\frac{37}{30},\frac{\mathrm{viii}}{\mathrm{xv}}\right)$

45.

$\left(\frac{10}{123},\mathrm{-1},\frac{2}{5}\right)$

47.

$\frac{1}{2}\left[\begin{array}{rrrr}\hfill \mathrm{ii}& \hfill \mathrm{ane}& \hfill -\mathrm{one}& \hfill -\mathrm{i}\\ \hfill 0& \hfill 1& \hfill 1& \hfill -1\\ \hfill 0& \hfill -1& \hfill 1& \hfill 1\\ \hfill 0& \hfill 1& \hfill -1& \hfill 1\end{array}\right]$

49.

$\frac{\mathrm{one}}{39}\left[\begin{array}{rrrr}\hfill \mathrm{iii}& \hfill 2& \hfill \mathrm{i}& \hfill -7\\ \hfill 18& \hfill -53& \hfill 32& \hfill 10\\ \hfill 24& \hfill -36& \hfill 21& \hfill \mathrm{nine}\\ \hfill -\mathrm{ix}& \hfill 46& \hfill -16& \hfill -5\end{array}\right]$

51.

$\left[\begin{array}{rrrrrr}\hfill \mathrm{i}& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill \mathrm{ane}& \hfill 0\\ \hfill -1& \hfill -\mathrm{one}& \hfill -\mathrm{one}& \hfill -\mathrm{one}& \hfill -1& \hfill \mathrm{i}\end{array}\right]$

55.

50% oranges, 25% bananas, xx% apples

57.

x harbinger hats, 50 beanies, forty cowboy hats

59.

Tom ate 6, Joe ate 3, and Albert ate 3.

61.

124 oranges, x lemons, viii pomegranates

11.8 Section Exercises

one.

A determinant is the sum and products of the entries in the matrix, so you tin ever evaluate that product—fifty-fifty if it does end up being 0.

three.

The inverse does not exist.

27.

$\left(\frac{\mathrm{ane}}{2},\frac{1}{3}\right)$

31.

$\left(-1,-\frac{\mathrm{ane}}{3}\right)$

37.

$\left(-\mathrm{ane},0,3\right)$

39.

$\left(\frac{1}{2},\mathrm{i},2\right)$

53.

$seven,000 in first account, $3,000 in second account.

55.

120 children, 1,080 adult

57.

4 gal yellow, vi gal bluish

59.

13 green tomatoes, 17 red tomatoes

61.

Strawberries 18%, oranges 9%, kiwi 10%

63.

100 for pic i, 230 for movie two, 312 for moving-picture show 3

65.

20–29: ii,100, thirty–39: two,600, 40–49: 825

67.

300 almonds, 400 cranberries, 300 cashews

Review Exercises

9.

$(300,\mathrm{threescore},000)$

15.

$\left(-1,-2,\mathrm{iii}\right)$

17.

$\left(x,\frac{8x}{\mathrm{v}},\frac{14\mathrm{10}}{\mathrm{five}}\right)$

21.

$\left(2,-3\right),\left(\mathrm{iii},2\right)$

31.

$\frac{2}{x+\mathrm{ii}},\frac{-\mathrm{iv}}{x+1}$

33.

$\frac{\mathrm{vii}}{x+5},\frac{-\mathrm{xv}}{{(\mathrm{10}+5)}^2}$

35.

$\frac{3}{x-5},\frac{-\mathrm{four}x+1}{{\mathrm{10}}^2+\mathrm{five}x+25}$

37.

$\frac{x-4}{(x^2-\mathrm{two})},\frac{5x+3}{{({\mathrm{ten}}^2-2)}^{\mathrm{two}}}$

39.

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

41.

undefined; dimensions do non friction match

43.

undefined; inner dimensions exercise not match

45.

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

47.

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

49.

undefined; inner dimensions do not match

51.

$\begin{array}{l}x-3z=7\\ y+2z=-\mathrm{five}\end{array}$ with infinite solutions

53.

$\left[\begin{array}{rrr}\hfill -2& \hfill \mathrm{ii}& \hfill \mathrm{i}\\ \hfill 2& \hfill -8& \hfill \mathrm{five}\\ \hfill \mathrm{xix}& \hfill -10& \hfill 22\end{array}|\begin{array}{r}\hfill \mathrm{seven}\\ \hfill 0\\ \hfill \mathrm{three}\end{array}\right]$

55.

$\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 3\\ \hfill \mathrm{-one}& \hfill 4& \hfill 0\\ \hfill 0& \hfill 1& \hfill 2\end{array}|\begin{array}{r}\hfill 12\\ \hfill 0\\ \hfill \mathrm{-7}\end{array}\right]$

61.

$\frac{1}{8}\left[\begin{array}{cc}\mathrm{two}& \mathrm{vii}\\ 6& 1\end{array}\right]$

65.

$\left(-\mathrm{twenty},40\right)$

67.

$\left(-1,0.2,0.3\right)$

69.

17% oranges, 34% bananas, 39% apples

75.

$\left(\mathrm{half-dozen},\frac{\mathrm{i}}{\mathrm{ii}}\right)$

79.

$\left(0,0,-\frac{1}{2}\right)$

Do Examination

five.

$\frac{\mathrm{ane}}{20}\left(10,5,4\right)$

7.

$\left(x,\frac{16\mathrm{ten}}{5}-\frac{13x}{\mathrm{v}}\right)$

nine.

$(-2\sqrt{2},-\sqrt{17}),\left(-2\sqrt{2},\sqrt{17}\right),\left(\mathrm{ii}\sqrt{2},-\sqrt{17}\right),\left(2\sqrt{2},\sqrt{17}\right)$

13.

$\frac{\mathrm{five}}{3x+1}-\frac{\mathrm{two}x+3}{{(3x+\mathrm{ane})}^{\mathrm{two}}}$

xv.

$\left[\begin{array}{cc}17& 51\\ -8& \mathrm{eleven}\end{array}\right]$

17.

$\left[\begin{array}{cc}12& -20\\ -\mathrm{fifteen}& 30\end{array}\right]$

21.

$\left[\begin{array}{rrr}\hfill 14& \hfill -2& \hfill 13\\ \hfill -2& \hfill 3& \hfill -\mathrm{six}\\ \hfill 1& \hfill -5& \hfill 12\end{array}|\begin{array}{r}\hfill 140\\ \hfill -1\\ \hfill \mathrm{eleven}\end{array}\right]$

25.

$\left(100,90\right)$

27.

$\left(\frac{1}{100},0\right)$

29.

32 or more cell phones per day

1-4 Exercises Guided Practice Geometry,

Source: https://openstax.org/books/algebra-and-trigonometry/pages/chapter-11

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