Chapter 8

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{aligned} -2 x+y &=-5 \\ x^{2}+y^{2} &=25 \end{aligned}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} -3 & -2 \\ 4 & 2 \end{array}\right]$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} 4 x-3 y-2 z &=21 \\ 6 y-5 z &=-8 \\ z &=-2 \end{aligned}\right.$$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$\left[\begin{array}{rrr} 1.3 & 0.2 & 3.2 \\ 0.2 & 6.2 & 0.2 \\ -0.4 & 4.4 & 0.3 \end{array}\right]$$

Use a determinant to determine whether the points are collinear. \(\left(2,-\frac{1}{2}\right),(-4,4),(6,-3)\)

Write the augmented matrix for the system of linear equations. What is the dimension of the augmented matrix? $$\left\\{\begin{aligned} x+10 y-2 z &=2 \\ 5 x-3 y+4 z &=0 \\ 2 x+y &=6 \end{aligned}\right.$$

Solve the system by the method of elimination and check any solutions algebraically. \(\left\\{\begin{array}{l}2 x+3 y=18 \\ 5 x-y=11\end{array}\right.\)

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{array}{r} 3 x+y=2 \\ x^{3}-2+y=0 \end{array}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 8 & -1 \\ 2 & 3 \\ -4 & 5 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 6 \\ -1 & -5 \\ 1 & 10 \end{array}\right]$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} 2 x+y-3 z &=10 \\ y+z &=12 \\ z &=2 \end{aligned}\right.$$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$\left[\begin{array}{rrr} 5.1 & 0.2 & 7.3 \\ -6.3 & 0.2 & 0.2 \\ 0.5 & 3.4 & 0.4 \end{array}\right]$$

Use a determinant to determine whether the points are collinear. \(\left(0, \frac{1}{2}\right),(2,-1),\left(-4, \frac{7}{2}\right)\)

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{rr} -7 & 33 \\ 4 & -19 \end{array}\right]$$

Write the augmented matrix for the system of linear equations. What is the dimension of the augmented matrix? $$\left\\{\begin{aligned} x-3 y+z &=1 \\ 4 y &=0 \\ 7 z &=-5 \end{aligned}\right.$$

Solve the system by the method of elimination and check any solutions algebraically. \(\left\\{\begin{array}{c}x+7 y=12 \\ 3 x-5 y=10\end{array}\right.\)

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{array}{r} x+y=0 \\ x^{3}-5 x-y=0 \end{array}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rrr} 1 & -1 & 3 \\ 0 & 6 & 9 \end{array}\right], \quad B=\left[\begin{array}{rrr} -2 & 0 & -5 \\ -3 & 4 & -7 \end{array}\right]$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} x-y+2 z &=22 \\ 3 y-8 z &=-9 \\ z &=-3 \end{aligned}\right.$$

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rr}3 & 4 \\\2 & -5\end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} 3 r+2 s=-6 \\ 2 r+6 s=3 \end{array}\right.$$

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{array}{l} -\frac{7}{2} x-y=-18 \\ 8 x^{2}-2 y^{3}=0 \end{array}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$\begin{aligned} &A=\left[\begin{array}{lllll} 4 & 5 & -1 & 3 & 4 \\ 1 & 2 & -2 & -1 & 0 \end{array}\right]\\\ &B=\left[\begin{array}{rrrrr} 1 & 0 & -1 & 1 & 0 \\ -6 & 8 & 2 & -3 & -7 \end{array}\right] \end{aligned}$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} 4 x-2 y+z &=8 \\ -y+z &=4 \\ z &=11 \end{aligned}\right.$$

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rr}11 & 6 \\\\-3 & 2\end{array}\right]$$

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} 1 & 2 & 2 \\ 3 & 7 & 9 \\ -1 & -4 & -7 \end{array}\right]$$

Write the augmented matrix for the system of linear equations. What is the dimension of the augmented matrix? $$\left\\{\begin{aligned} 9 x+2 y-3 z &=20 \\ -25 y+11 z &=-5 \end{aligned}\right.$$

Solve the system by the method of elimination and check any solutions algebraically. $\underline{\phantom{xxx}}\left\\{\begin{array}{c} 8 r+16 s=20 \\ 16 r+50 s=55 \end{array}\right.$$

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{array}{l} y=x^{3}-3 x^{2}+4 \\ y=-2 x+4 \end{array}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rrr} -1 & 4 & 0 \\ 3 & -2 & 2 \\ 5 & 4 & -1 \\ 0 & 8 & -6 \\ -4 & -1 & 0 \end{array}\right], B=\left[\begin{array}{rrr} -3 & 5 & 1 \\ 2 & -4 & -7 \\ 10 & -9 & -1 \\ 3 & 2 & -4 \\ 0 & 1 & -2 \end{array}\right]$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} 5 x-8 z &=22 \\ 3 y-5 z &=10 \\ z &=-4 \end{aligned}\right.$$

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rrr}-4 & 6 & 3 \\\7 & -2 & 8 \\\1 & 0 & -5\end{array}\right]$$

Use Cramer's Rule to solve (if possible) the system of equations. \(\left\\{\begin{array}{r}-7 x+11 y=-1 \\ 3 x-9 y=9\end{array}\right.\)

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{lll} 1 & 1 & 1 \\ 3 & 5 & 4 \\ 3 & 6 & 5 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} 5 u+6 v=24 \\ 3 u+5 v=18 \end{array}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} x+y &=0 \\ 4 x+3 y &=10 \end{aligned}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rrr} 6 & 0 & 3 \\ -1 & -4 & 0 \end{array}\right], \quad B=\left[\begin{array}{ll} 8 & -1 \\ 4 & -3 \end{array}\right]$$

Perform the row operation and write the equivalent system. What did the operation accomplish? Add Equation 1 to Equation 2. $$\left\\{\begin{aligned} x-2 y+3 z &=5 \\ -x+3 y-5 z &=4 \\ 2 x &-3 z=0 \end{aligned}\right.$$ Equation 1 Equation 2 Equation 3

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rrr}-2 & 9 & 4 \\\7 & -6 & 0 \\\6 & 7 & -6\end{array}\right]$$

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{lll} 1 & 6 & 10 \\ 3 & 4 & 0 \\ 2 & 5 & 5 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{aligned} 3 u+11 v &=4 \\ -2 u-5 v &=9 \end{aligned}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{c} x+2 y=1 \\ 5 x-4 y=-23 \end{array}\right.$$

Perform the row operation and write the equivalent system. What did the operation accomplish? Add -2 times Equation 1 to Equation 3. $$\left\\{\begin{aligned} x-2 y+3 z &=5 \\ -x+3 y-5 z &=4 \\ 2 x &-3 z=0 \end{aligned}\right.$$ Equation 1 Equation 2 Equation 3

Find the determinant of the matrix. Expand by cofactors on each indicated row or column. \(\left[\begin{array}{rrr}-3 & 2 & 1 \\ 4 & 5 & 6 \\ 2 & -3 & 1\end{array}\right]\) (a) Row 1 (b) Column 2

Use Cramer's Rule to solve (if possible) the system of equations. \(\left\\{\begin{array}{l}4 x-3 y=-10 \\ 6 x+9 y=12\end{array}\right.\)

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} 1 & 1 & 2 \\ 3 & 1 & 0 \\ -2 & 0 & 3 \end{array}\right]$$

Write the system of linear equations represented by the augmented matrix. (Use the variables \(x, y, z,\) and \(w,\) if applicable.) $$\left[\begin{array}{rrrrr} 0 & 12 & 3 & \vdots & 0 \\ -2 & 18 & 0 & \vdots & 10 \\ 1 & 7 & -8 & \vdots & 43 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{c} -6 x+5 y=-15 \\ 4 x+12 y=10 \end{array}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{l} 2 x-y+2=0 \\ 4 x+y-5=0 \end{array}\right.$$

Evaluating an Expression Evaluate the expression. $$\left[\begin{array}{rr} -5 & 0 \\ 3 & -6 \end{array}\right]+\left[\begin{array}{rr} 7 & 1 \\ -2 & -1 \end{array}\right]+\left[\begin{array}{rr} -10 & -8 \\ 14 & 6 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{aligned} x+y+z &=6 \\ 2 x-y+z &=3 \\ 3 x &-z=0 \end{aligned}\right.$$