Systems of Equations (I) - Ch. 7.2 & 7.3

I. Systems of Equations (Ch. 7-287-3) I. DEFINITION system of equations- a group of conditional equations a solution to the system is defined by the intersection (^) of the equations II. DIFFERENT TYPES OF SOLUTIONS: if a solution or solutions exist, the system is considered consistent subsequently, if the number of solutions is finite, the system is independent if the system has infinitely many solutions, it is dependent if no solution exists, the system is inconsistent *** no solutions one solution infinite solutions inconsistent consistent, independent consistent, dependent Solve and classify each system of equations 1. 2. 3 \[\begin{cases}x=6\\ 6x+5y=6\end{cases}\] \[\{\begin{matrix}3x-2y=\\ x+y=4\end{matrix}\}\] \[\{\begin{matrix}y=\frac{2}{3}x-\frac{5}{3}\\ 6x-9y=3\end{matrix}\] \[6(6)\cdot5y=6\] \[\begin{cases}3x-2y=7\\ 2x+2y=8\end{cases}\] \[6x-9(\frac{3}{3}x-\frac{5}{3})=3\] \[36+5y=6\] \[6x-6x+15=3\] \[5y=-30\] \[15\div3\] \[\overline{4}=-6\] \[x=3\] \[3+y=4\] \[y=1\] \[(6,\cdot6)\] inconsistent consistentindependent \[(3,1)\] consistentindependent 4. Seth has $8.80 in pennies and nickels. He has a total of 240 coins. Find out how many of each coin Seth has. \[\{\begin{matrix}p+n=240\\ 0~01~p\cdot0.05n=8.80\end{matrix}\] \[\{\begin{matrix}p+n=240\\ p+5n=880\end{matrix}\] \[4n=640\] \[n=160\] 80 pennies 160 nickels \[p+160=240\] True or False? (3, -2, 1) is a solution to {x + y + z = 2 {6x - 4y + 5z = 31 {5x + 2y + 2z = 13 (3) + (-2) + (1) = 2 6(3) - 4(-2) - 5(1) = 31 5(3) + 2(-2) + 2(1) = 13 QED 18 + 8 + 5 = 31 15 - 4 + 2 = 13 2 = 2 31 = 31 13 = 13 True {x + y + z = 3 {3x + y + 2z = 0 3(2) + y + 2z = 0 {4y + z = 3 y + 2z = -6 {4 + 2z = 10 {4 + 2z = 10 {3y - 6z = -6 (2, 0, -3) consistent, independent (0) - 2 - 3 z = -3 {x + y = 4 {-x - z = 9 {2x + z = 0 {x - z = 9 (9) + y = 4 2(9) + z = 0 (9, -5, -18) {2x + z = 0 y = -5 18 + z = 0 consistent, independent x = 9 z = -18 x - 3y + z = 4 -x + 2y - 5z = 3 5x - 13y + 13z = 8 {x - 3y + z = 4 {-x + 2y - 5z = 3 {-y - 4z = 7 {5x - 10y + 25z + 15 {5x - 13y + 13z + 8 {-3y - 12z + 23 3y + 12z = 21 -3y - 12z = 23 0 ≠ 2 inconsistent {2x + y - 2z = -1 {3x - 3y - z = 5 {x - 2y + 3z = 6 {6x + 3y - 6z = -3 {3x - 3y - z = 5 {9x - 7z = 2 5(0) - 2(4) + 3(2) = 6 -3 = 6 x - 2(4) + 3(2) = 6 -5 ≠ 6 5x - z = 4 x = 1 consistent, independent (1, -1, 1) 10. Find the equation of the parabola y = ax^2 + bx + c that passes through the following three points: (2, -40), (1, 1), (3, 15) (-40) = a(2)^2 + b(2) + c 4a + 2b + c = -40 a + b + c = 7 9a + 3b + c = 15 4a + 2b + c = -40 -2a - 2b - 2c = -14 2a - c = -24 2(51) - c = -54 102 - c = -54 c = 156 51 + b + 156 = 7 b = -200 y = 51x^2 - 200x + 156 11. 2x - y + 2z = 15 x + y + z = 3 3x - y + 2z = 18 2x - y + 2z = 15 -x + y + z = 3 x + 3z = 18 2x + 3z = 21 3 + 3z = 18 z = 5 x + 3z = 18 x = 3 -3 + y + 5 = 3 y = 1 (3, 1, 5) consistent, independent