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# Solving Systems

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

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)