11.7E: Solving Systems with Inverses (Exercises)
For the following exercises, find the inverse of the matrix.
61. \(\left[\begin{array}{rr}-0.2 & 1.4 \\ 1.2 & -0.4\end{array}\right]\)
62. \(\left[\begin{array}{rr}\frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{4} & \frac{3}{4}\end{array}\right]\)
63. \(\left[\begin{array}{ccc}12 & 9 & -6 \\ -1 & 3 & 2 \\ -4 & -3 & 2\end{array}\right]\)
64. \(\left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 2 & 3 \\ 3 & 2 & 1\end{array}\right]\)
For the following exercises, find the solutions by computing the inverse of the matrix.
65.
\(0.3 x-0.1 y=-10\)
\(-0.1 x+0.3 y=14\)
66.
\(0.4 x-0.2 y=-0.6\)
\(-0.1 x+0.05 y=0.3\)
67.
\begin{array}{r}
4 x+3 y-3 z=-4.3 \\
5 x-4 y-z=-6.1 \\
x+z=-0.7
\end{array}
68
\(-2x - 3y _2 z=3\)
\(-x+2 y+4 z=-5\)
\(-2 y+5 z=-3\)
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
69. Students were asked to bring their favorite fruit to class. \(90 \%\) of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were \(5 \%\) more popular than bananas, what are the percentages of each individual fruit?
70. A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at \(\$ 2\) and the chocolate chip cookies at \(\$ 1 .\) They raised \(\$ 250\) and sold 175 items. How many brownies and how many cookies were sold? \(?\)