Chapter 6

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{array}{r} 2 x+y=3 \\ -2 x-y=4 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}-1 & 0 & -2 \\\4 & -2 & 1\end{array}\right], \quad B=\left[\begin{array}{rr}2 & -2 \\\5 & -1 \\\0 & 1\end{array}\right]$$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{rrr} 2 & -3 & 1 \\ 5 & -6 & 3 \\ 3 & 2 & 0 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{r} x+y+z=3 \\ x+y+2 z=4 \\ 2 x+2 y+3 z=7 \end{array} $$

Geometry The perimeter of a triangle is 105 inches. The longest side is 22 inches longer than the shortest side. The sum of the lengths of the two shorter sides is 15 inches more than the length of the longest side. Find the lengths of the sides of the triangle.

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. (GRAPH CAN'T COPY) $$ \begin{array}{c} x-4 y=4 \\ 2 x-8 y=4 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}1 & -1 & 0 \\\2 & -1 & 5 \\\6 & 1 & -4\end{array}\right], \quad B=\left[\begin{array}{rrr}-1 & 3 & -1 \\\7 & -7 & 1\end{array}\right]$$

Use the concept of the area of a triangle to determine if the three points are collinear. $$ (1,3),(-3,11),(2,1) $$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{rrrr} 1 & -1 & 0 & 0 \\ -1 & 5 & -1 & 0 \\ 0 & -1 & 5 & -1 \\ 0 & 0 & -1 & 1 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{rr} -x+2 y+4 z= & 10 \\ 3 x-2 y-2 z= & -12 \\ x+2 y+6 z= & 8 \end{array} $$

A sum of \(\$ 20,000\) is invested in three mutual funds. In one year the first fund grew by \(5 \%,\) the second by \(7 \%,\) and the third by \(10 \% .\) Total earnings for the year were \(\$ 1650\). The amount invested in the third fund was 4 times the amount invested in the first fund. Find the amount invested in each fund.

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. (GRAPH CAN'T COPY) $$ \begin{array}{r} 3 x-y=7 \\ -2 x+y=-5 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}2 & -1 & -5 \\\4 & -1 & 6 \\\\-2 & 0 & 9 \end{array}\right], \quad B=\left[\begin{array}{rr}1 & 2 \\\\-1 & -1 \\\2 & 0\end{array}\right]$$

Use the concept of the area of a triangle to determine if the three points are collinear. $$ (3,6),(-1,-6),(5,11) $$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{llll} 3 & 1 & 0 & 0 \\ 1 & 3 & 1 & 0 \\ 0 & 1 & 3 & 1 \\ 0 & 0 & 1 & 3 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{l} 4 x-2 y+4 z=8 \\ 3 x-7 y+6 z=4 \\ -x-5 y+2 z=7 \end{array} $$

Home Prices Prices of homes can depend on several factors such as size and age. The table shows the selling prices for three homes. In this table, price \(P\) is given in thousands of dollars, age \(A\) in years, and home size \(S\) in thousands of square feet. These data may be modeled by \(P=a+b A+c S\) $$ \begin{array}{ccc} \hline \text { Price (P) } & \text { Age (A) } & \text { Size (S) } \\ \hline 190 & 20 & 2 \\ 320 & 5 & 3 \\ 50 & 40 & 1 \end{array} $$ (a) Write a system of linear equations whose solution gives \(a, b,\) and \(c\) (b) Solve this system of linear equations. (c) Predict the price of a home that is 10 years old and has 2500 square feet.

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{array}{l} -x+2 y=3 \\ 3 x-y=1 \end{array} $$

Use the concept of the area of a triangle to determine if the three points are collinear. $$ (-2,-5),(4,4),(2,3) $$

Represent the system of linear equations in the form \(A X=B\) \(2 x-3 y=7\) \(-3 x-4 y=9\)

Solve the system, if possible. $$ \begin{array}{r} x-y+z=1 \\ x+2 y-z=2 \\ y-z=0 \end{array} $$

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{array}{rr} x-2 y= & -6 \\ -2 x+y= & 6 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rr}3 & -1 \\\2 & -2 \\\0 & 4\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & -4 & 0 \\\\-1 & 3 & 2\end{array}\right]$$

Use the concept of the area of a triangle to determine if the three points are collinear. $$ (4,-5),(-2,10),(6,-10) $$

Represent the system of linear equations in the form \(A X=B\) \(-x+3 y=10\) \(2 x-6 y=-1\)

Solve the system, if possible. $$ \begin{array}{rr} x-y-2 z= & -11 \\ x-2 y-z= & -11 \\ -x+y+3 z= & 14 \end{array} $$

Business Production A business has three machines that manufacture containers. Together they can make 100 containers per day, whereas the two fastest machines can make 80 containers per day. The fastest machine makes 34 more containers per day than the slowest machine. (a) Let \(x, y,\) and \(z\) be the numbers of containers that the machines make from fastest to slowest. Write a system of three equations whose solution gives the number of containers each machine can make. (b) Solve the system of equations.

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{array}{r} 2 x-3 y=1 \\ x+y=-2 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}2 & -1 & 3 \\\0 & 1 & 0 \\\2 & -2 & 3\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & 5 & -1 \\\0 & 1 & 3 \\\\-1 & 2 & 1\end{array}\right]$$

Shade the region of feasible solutions for the following constraints. $$ \begin{aligned} &x+y \leq 4\\\ &x+y \geq 1\\\ &x \geq 0, y \geq 0 \end{aligned} $$

If a line passes through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) then an equation of this line can be found by calculating the determinant. $$ \operatorname{det}\left[\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right]=0 $$ Find the standard form ax \(+b y=c\) of the line passing through the given points. $$ (2,1) \text { and }(-1,4) $$

Represent the system of linear equations in the form \(A X=B\) \(\frac{1}{2} x-\frac{3}{2} y=\frac{1}{4}\) \(-x+2 y=5\)

Solve the system, if possible. $$ \begin{aligned} 2 x-4 y+2 z &=11 \\ x+3 y-2 z &=-9 \\ 4 x-2 y+z &=7 \end{aligned} $$

When using elimination and substitution, explain how to recognize a system of linear equations that has no solutions.

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{array}{rr} 2 x-y= & -4 \\ -4 x+2 y= & 8 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}1 & -2 & 5 \\\1 & 0 & -2 \\\1 & 3 & 2\end{array}\right], \quad B=\left[\begin{array}{rrr}-1 & 4 & 2 \\\\-3 & 0 & 1 \\\5 & 1 & 0\end{array}\right]$$

Shade the region of feasible solutions for the following constraints. $$ \begin{array}{l} x+2 y \leq 8 \\ 2 x+y \geq 2 \\ x \geq 0, y \geq 0 \end{array} $$

If a line passes through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) then an equation of this line can be found by calculating the determinant. $$ \operatorname{det}\left[\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right]=0 $$ Find the standard form ax \(+b y=c\) of the line passing through the given points. $$ (-1,3) \text { and }(4,2) $$

Represent the system of linear equations in the form \(A X=B\) \(-1.1 x+3.2 y=-2.7\) \(5.6 x-3.8 y=-3.0\)

Solve the system, if possible. $$ \begin{array} x-4 y+z= 9 \\ 3 y-2 z= -7 \\ -x & +z=0 \end{array} $$

When using elimination and substitution, explain how to recognize a system of linear equations that has infinitely many solutions.

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{aligned} 3 x-y &=-2 \\ -3 x+y &=2 \end{aligned} $$

Shade the region of feasible solutions for the following constraints. $$ \begin{aligned} &3 x+2 y \leq 12\\\ &2 x+3 y \leq 12\\\ &x \geq 0, y \geq 0 \end{aligned} $$

If a line passes through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) then an equation of this line can be found by calculating the determinant. $$ \operatorname{det}\left[\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right]=0 $$ Find the standard form ax \(+b y=c\) of the line passing through the given points. $$ (6,-7) \text { and }(4,-3) $$

Represent the system of linear equations in the form \(A X=B\) \(x-2 y+z=5\) \(3 y-z=6\) \(5 x-4 y-7 z=0\)

Solve the system, if possible. $$ \begin{aligned} 2 x-y-z &=0 \\ x-y-z &=-2 \\ 3 x-2 y-2 z &=-2 \end{aligned} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{array}{r} x+2 y=0 \\ 3 x+7 y=1 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{llll}5 & -3\end{array}\right], \quad \quad \quad \quad \quad B=\left[\begin{array}{l}1 \\\3\end{array}\right]$$

Shade the region of feasible solutions for the following constraints. $$ \begin{aligned} &x+y \leq 4\\\ &x+4 y \geq 4\\\ &x \geq 0, y \geq 0 \end{aligned} $$

If a line passes through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) then an equation of this line can be found by calculating the determinant. $$ \operatorname{det}\left[\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right]=0 $$ Find the standard form ax \(+b y=c\) of the line passing through the given points. \((5,1)\) and \((2,-2)\)