Chapter 8

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{c} x\quad+4 z=1 \\ x+y+10 z=10 \\ 2 x-y+2 z=-5 \end{array}\right.$$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{llll} 1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 8 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$\left|\begin{array}{rrrrrr} -2 & 0 & 1 & 4 & 3 & -2 \\ -3 & 3 & 0 & -2 & 1 & -1 \\ 4 & 5 & -1 & 0 & 7 & 3 \\ 2 & 4 & 3 & 2 & 0 & 1 \\ 1 & 3 & 4 & 2 & -4 & 0 \\ -5 & 2 & -1 & 3 & 2 & -3 \end{array}\right|$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{aligned} \frac{2 x+5}{2}+\frac{y-1}{3} &=-1 \\ 2 x-y &=12 \end{aligned}\right.\)

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{l} y=-x \\ y=x^{3}+3 x^{2}+2 x \end{array}\right.$$

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{rrr} 5 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & 7 \end{array}\right], \quad B=\left[\begin{array}{rrr} \frac{1}{5} & 0 & 0 \\ 0 & -\frac{1}{8} & 0 \\ 0 & 0 & \frac{1}{2} \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{rr} x-2 y+z= & 2 \\ 2 x+2 y-3 z= & -4 \\ 5 x\quad+z= & 1 \end{array}\right.$$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{cccc} 1 & 0 & 2 & 1 \\ 0 & 1 & -3 & 10 \\ 0 & 0 & 1 & 0 \end{array}\right]$$

Find \((a)|A|,(b)|B|,(c) A B,\) and \((d)|A B| .\) What do you notice about \(|A B| ?\) $$A=\left[\begin{array}{rr}-2 & 1 \\\4 & -2\end{array}\right], \quad B=\left[\begin{array}{rr}1 & 2 \\\0 & -1\end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}-5 x+6 y=-3 \\ 20 x-24 y=12\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{array}{r} -2 x+y=7 \\ x+3 y=0 \end{array}\right.$$

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{r} 5 \\ -3 \\ 4 \end{array}\right], \quad B=\left[\begin{array}{lll} 2 & -8 & 4 \end{array}\right]$$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{lllr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 2 \end{array}\right]$$

At this point in the text, you have learned several methods for finding an equation of a line that passes through two given points. Briefly describe these methods and discuss the advantages and disadvantages of each.

Find \((a)|A|,(b)|B|,(c) A B,\) and \((d)|A B| .\) What do you notice about \(|A B| ?\) $$A=\left[\begin{array}{rr}4 & 0 \\\3 & -2\end{array}\right], \quad B=\left[\begin{array}{ll}-1 & 1 \\\\-2 & 2\end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{r}7 x+8 y=16 \\ -14 x-16 y=-4\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{aligned} x+y &=8 \\ 4 x+4 y &=0 \end{aligned}\right.$$

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{l} 5 \\ 6 \end{array}\right], \quad B=\left[\begin{array}{llll} -3 & -1 & -5 & -9 \end{array}\right]$$

Write the matrix in row-echelon form. Remember that the row-echelon form of a matrix is not unique. $$\left[\begin{array}{rrr} 1 & -4 & 5 \\ -2 & 6 & -6 \end{array}\right]$$

Find the general form of the equation of the line that passes through the two points. \((-1,5),(7,3)\)

Find \((a)|A|,(b)|B|,(c) A B,\) and \((d)|A B| .\) What do you notice about \(|A B| ?\) $$A=\left[\begin{array}{rrr}3 & 2 & 0 \\\\-1 & -3 & 4 \\\\-2 & 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{rrr}-3 & 0 & 1 \\\0 & 2 & -1 \\\\-2 & -1 & 1\end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{r}2.5 x-3 y=1.5 \\ x-1.2 y=-3.6\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{aligned} x-2 y &=-3 \\ 5 x+6 y &=17 \end{aligned}\right.$$

Operations with Matrices Find, if possible, (a) \(A B,\) (b) \(B A,\) and (c) \(A^{2}\). (Note: \(A^{2}=A A\).) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ -1 & 8 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{l} 12 x+5 y+z=0 \\ 23 x+4 y-z=0 \end{array}\right.$$

Write the matrix in row-echelon form. Remember that the row-echelon form of a matrix is not unique. $$\left[\begin{array}{ccc} 1 & -3 & 2 \\ 5 & 0 & 7 \end{array}\right]$$

Find the general form of the equation of the line that passes through the two points. \((0,-6),(-2,10)\)

Find \((a)|A|,(b)|B|,(c) A B,\) and \((d)|A B| .\) What do you notice about \(|A B| ?\) $$A=\left[\begin{array}{rrr}2 & 0 & 1 \\ 1 & -4 & 2 \\\3 & 1 & 0\end{array}\right], \quad B=\left[\begin{array}{rrr}2 & -1 & 4 \\\0 & 1 & 0 \\\3 & -2 & 1\end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}6.3 x+7.2 y=5.4 \\ 5.6 x+6.4 y=4.8\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{array}{r} -5 x+2 y=-2 \\ x-2 y=6 \end{array}\right.$$

Operations with Matrices Find, if possible, (a) \(A B,\) (b) \(B A,\) and (c) \(A^{2}\). (Note: \(A^{2}=A A\).) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 6 & 3 \\ -2 & -4 \end{array}\right], \quad B=\left[\begin{array}{rr} -2 & 0 \\ 2 & 4 \end{array}\right]$$

Write the matrix in row-echelon form. Remember that the row-echelon form of a matrix is not unique. $$\left[\begin{array}{rrrr} 1 & 2 & -1 & 3 \\ 3 & 7 & -5 & 14 \\ -2 & -1 & -3 & 8 \end{array}\right]$$

Find the general form of the equation of the line that passes through the two points. \((3,-3),(10,-1)\)

Use the matrix capabilities of a graphing utility to find (a) \(|A|,\) (b) \(|B|,(c) A B,\) and \((d)|A B| .\) What do you notice about \(|A B| ?\) $$A=\left[\begin{array}{rrrr} 6 & 4 & 0 & 1 \\ 2 & -3 & -2 & -4 \\ 0 & 1 & 5 & 0 \\ -1 & 0 & -1 & 1 \end{array}\right], \quad B=\left[\begin{array}{rrrr} 0 & -5 & 0 & -2 \\ -2 & 4 & -1 & -4 \\ 3 & 0 & 1 & 0 \\ 1 & -2 & 3 & 0 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}0.2 x-0.5 y=-27.8 \\ 0.3 x+0.4 y=68.7\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{array}{r} x^{2}+y=-1 \\ -x+2 y=5 \end{array}\right.$$

Operations with Matrices Find, if possible, (a) \(A B,\) (b) \(B A,\) and (c) \(A^{2}\). (Note: \(A^{2}=A A\).) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 3 & -1 \\ 1 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & -3 \\ 3 & 1 \end{array}\right]$$

Find a system of linear equations that has the given solution. (There are many correct answers.) $$(3,-4,2)$$

Write the matrix in row-echelon form. Remember that the row-echelon form of a matrix is not unique. $$\left[\begin{array}{rrrr} 1 & -3 & 0 & -7 \\ -3 & 10 & 1 & 23 \\ 4 & -10 & 2 & -24 \end{array}\right]$$

Find the general form of the equation of the line that passes through the two points. \((-4,12),(4,2)\)

Use the matrix capabilities of a graphing utility to find (a) \(|A|,\) (b) \(|B|,(c) A B,\) and \((d)|A B| .\) What do you notice about \(|A B| ?\) $$A=\left[\begin{array}{rrrr} -1 & 5 & 2 & 0 \\ 0 & 0 & 1 & 1 \\ 3 & -3 & -1 & 0 \\ 4 & 2 & 4 & -1 \end{array}\right], \quad B=\left[\begin{array}{rrrr} 1 & 5 & 0 & 0 \\ 10 & -1 & 2 & 4 \\ 2 & 0 & 0 & 1 \\ -3 & 2 & 5 & 0 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{r}0.2 x+0.6 y=-1 \\ x-0.5 y=2\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{array}{c} x^{2}-y=3 \\ x-y=1 \end{array}\right.$$

Operations with Matrices Find, if possible, (a) \(A B,\) (b) \(B A,\) and (c) \(A^{2}\). (Note: \(A^{2}=A A\).) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 3 \\ -3 & 1 \end{array}\right]$$

Find a system of linear equations that has the given solution. (There are many correct answers.) $$(-5,-2,1)$$

Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. $$\left[\begin{array}{rrr} 3 & 3 & 3 \\ -1 & 0 & -4 \\ 2 & 4 & -2 \end{array}\right]$$

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=-\left|\begin{array}{ll} y & z \\ w & x \end{array}\right|$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 3 x+4 y=-2 \\ 5 x+3 y=4 \end{array}\right.$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}\frac{1}{x}+\frac{3}{y}=2 \\\ \frac{4}{x}-\frac{1}{y}=-5\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{aligned} -x-y &=3 \\ x^{2}+y^{2}-4 x-21 &=0 \end{aligned}\right.$$