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# Solving Systems

### Solving Systems of Linear Equations by Addition and in Three Variables

Question | Answer |
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The method of solving systems of linear equations, in which the two equations are added together to eliminate one of the variables, called "....." | Addition Method if A=B and C=D => A+C=B+D |

Solve the following system using addition method x-y=2 x+y=8 | (x-y=2) + ( x+y=8) 1)2x=10 2) x=5 3) x+y=8; 5+y=8; y=8-5; y=3 Check. 5-3=2; 2=2 true 5+3=8: 8=8 true (5,3) - solution |

Solve the following system using addition method x-2y=11 3x-y=13 | Multiply first equation by (-3) 1) x-2y=11 2) (-3)*(x-2y)=(-3)*11 3) -3x+6y=-33 4)add -3x+6y=-33 to 3x-y=13 5) 5y=-20 ; y=-4 6) x-2y=11 ; x=11+2y; x=11+2*(-4)=3 Check. 3-2*(-4)=11 true 3*3-(-4)=13 true (3,-4) - solution |

Solve the following system using addition method x-3y=5 2x-6y=-3 | Multiply first equation by (-2) 1) x-3y=5 2) (-2)*(x-3y)=(-2)*5 3) -2x+6y=-10 4) add -2x+6y=-10 to 2x-6y=-3 5) 0=-13 false This is an inconsistent system (two parallel lines) with no solution (with no intersection point). |

Solve the following system using addition method 4x-3y=5 -8x+6y=-10 | Multiply first equation by 2 1)4x-3y=5 2) 2*(4x-3y)=5*2 3) 8x -6y=10 4) add 8x -6y=10 to -8x+6y=-10 5) 0=0 true System has infinite number of solutions. Graphs of this equations are identical |

Solve the following system using addition method 4x+3y=14 3x-2y=2 | Multiply first equation by (-3) and second by 4 1) 4x+3y=14 2) (-3)*(4x+3y)=14*(-3) 3) -12x-9y=-42 4) 3x-2y=2 5) 4*(3x-2y)=2*4 6) 12x-8y=8 7) add -12x-9y=-42 to 12x-8y=8 8) -17y=-34; y=2 8) 4x+3y=14 ; 4x=14-3y; 4x=14-3*2; 4x=8; x=2 (2,2) - solution |

An ordered triple (x,y,z) is the solution of _______. | Linear equation in three variables |

The system is _____________ if a) Three planes have a single point in common. This point represents the single solution of the system. b) Three planes intersect at all points of a single line. This system has infinitely many solutions. | Consistent |

Three planes intersect at no point common to all three. This system has no solution. This system is ____________. | Inconsistent |

Three planes coincide at all points on the plane. This system is ________, and the equations are _________________. | Consistent, dependent |

Solve the following system (1) 3x+2y-z=0 (2) x-y+5z=2 (3) 2x+3y+3z=7 | Eq.(2)*(-3),add to eq.(1); (3x+2y-z=0)and(-3x-6y+3z=-6)-->eq.(4) -4y+2z=-6;Eq.(2)*(-2),add to eq.(3) ;(-2x+2y-10z=-4)and (2x+3y+3z=7)-->eq.(5) 5y-7z=3; Eq. (4)*5 and Eq.(5)*4;add (4)to(5) -18z=-18,z=-1 to(4) -4y+2*1=-6;y=2,to (2) x-2+5*1=2,x=-1 (-1,2,1) |