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Three-variable systems contain three equations with three unknowns. These systems can be solved using elimination, substitution, or matrices. The solution is an ordered triple that makes all three original equations true.
Solve each \(3\)X\(3\) system of linear equations.
\(\textbf{1)}\) \(2x+3y-5z=-7\)
\(\hspace{12pt}3x-6y+4z=3\)
\(\hspace{12pt}x+4y+2z=15\)
\(\hspace{33pt}\)The answer is \((1,2,3)\) or \(x=1,y=2,z=3\).\(\text{Multiply }x+4y+2z=15\text{ by }-2.\)
\(\,\,\,\,\,-2x-8y-4z=-30\)
\(\text{Add this to }2x+3y-5z=-7.\)
\(\,\,\,\,\,2x+3y-5z=-7\)
\(\,\,\,\,\,-2x-8y-4z=-30\)
\(\,\,\,\,\,-5y-9z=-37\)
\(\,\,\,\,\,5y+9z=37\)
\(\text{Multiply }x+4y+2z=15\text{ by }-3.\)
\(\,\,\,\,\,-3x-12y-6z=-45\)
\(\text{Add this to }3x-6y+4z=3.\)
\(\,\,\,\,\,3x-6y+4z=3\)
\(\,\,\,\,\,-3x-12y-6z=-45\)
\(\,\,\,\,\,-18y-2z=-42\)
\(\,\,\,\,\,9y+z=21\)
\(\text{Multiply }9y+z=21\text{ by }-9.\)
\(\,\,\,\,\,-81y-9z=-189\)
\(\text{Add this to }5y+9z=37.\)
\(\,\,\,\,\,-76y=-152\)
\(\,\,\,\,\,y=2\)
\(\,\,\,\,\,9(2)+z=21\)
\(\,\,\,\,\,z=3\)
\(\,\,\,\,\,x+4(2)+2(3)=15\)
\(\,\,\,\,\,x=1\)
\(\text{The solution is }(1,2,3).\)
\(\,\,\,\,\,-2x-8y-4z=-30\)
\(\text{Add this to }2x+3y-5z=-7.\)
\(\,\,\,\,\,2x+3y-5z=-7\)
\(\,\,\,\,\,-2x-8y-4z=-30\)
\(\,\,\,\,\,-5y-9z=-37\)
\(\,\,\,\,\,5y+9z=37\)
\(\text{Multiply }x+4y+2z=15\text{ by }-3.\)
\(\,\,\,\,\,-3x-12y-6z=-45\)
\(\text{Add this to }3x-6y+4z=3.\)
\(\,\,\,\,\,3x-6y+4z=3\)
\(\,\,\,\,\,-3x-12y-6z=-45\)
\(\,\,\,\,\,-18y-2z=-42\)
\(\,\,\,\,\,9y+z=21\)
\(\text{Multiply }9y+z=21\text{ by }-9.\)
\(\,\,\,\,\,-81y-9z=-189\)
\(\text{Add this to }5y+9z=37.\)
\(\,\,\,\,\,-76y=-152\)
\(\,\,\,\,\,y=2\)
\(\,\,\,\,\,9(2)+z=21\)
\(\,\,\,\,\,z=3\)
\(\,\,\,\,\,x+4(2)+2(3)=15\)
\(\,\,\,\,\,x=1\)
\(\text{The solution is }(1,2,3).\)
\(\textbf{2)}\) \(4x-2y+3z=20\)
\(\hspace{12pt}x-2y+3z=14\)
\(\hspace{12pt}4x+5y-8z=-24\)
\(\hspace{33pt}\)The answer is \((2,0,4)\) or \(x=2,y=0,z=4\).\(\text{Multiply }x-2y+3z=14\text{ by }-4.\)
\(\,\,\,\,\,-4x+8y-12z=-56\)
\(\text{Add this to }4x-2y+3z=20.\)
\(\,\,\,\,\,4x-2y+3z=20\)
\(\,\,\,\,\,-4x+8y-12z=-56\)
\(\,\,\,\,\,6y-9z=-36\)
\(\text{Multiply }4x-2y+3z=20\text{ by }-1.\)
\(\,\,\,\,\,-4x+2y-3z=-20\)
\(\text{Add this to }4x+5y-8z=-24.\)
\(\,\,\,\,\,4x+5y-8z=-24\)
\(\,\,\,\,\,-4x+2y-3z=-20\)
\(\,\,\,\,\,7y-11z=-44\)
\(\,\,\,\,\,-7y+11z=44\)
\(\text{Multiply }6y-9z=-36\text{ by }7.\)
\(\,\,\,\,\,42y-63z=-252\)
\(\text{Multiply }-7y+11z=44\text{ by }6.\)
\(\,\,\,\,\,-42y+66z=264\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,3z=12\)
\(\,\,\,\,\,z=4\)
\(\,\,\,\,\,6y-9(4)=-36\)
\(\,\,\,\,\,y=0\)
\(\,\,\,\,\,x-2(0)+3(4)=14\)
\(\,\,\,\,\,x=2\)
\(\text{The solution is }(2,0,4).\)
\(\,\,\,\,\,-4x+8y-12z=-56\)
\(\text{Add this to }4x-2y+3z=20.\)
\(\,\,\,\,\,4x-2y+3z=20\)
\(\,\,\,\,\,-4x+8y-12z=-56\)
\(\,\,\,\,\,6y-9z=-36\)
\(\text{Multiply }4x-2y+3z=20\text{ by }-1.\)
\(\,\,\,\,\,-4x+2y-3z=-20\)
\(\text{Add this to }4x+5y-8z=-24.\)
\(\,\,\,\,\,4x+5y-8z=-24\)
\(\,\,\,\,\,-4x+2y-3z=-20\)
\(\,\,\,\,\,7y-11z=-44\)
\(\,\,\,\,\,-7y+11z=44\)
\(\text{Multiply }6y-9z=-36\text{ by }7.\)
\(\,\,\,\,\,42y-63z=-252\)
\(\text{Multiply }-7y+11z=44\text{ by }6.\)
\(\,\,\,\,\,-42y+66z=264\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,3z=12\)
\(\,\,\,\,\,z=4\)
\(\,\,\,\,\,6y-9(4)=-36\)
\(\,\,\,\,\,y=0\)
\(\,\,\,\,\,x-2(0)+3(4)=14\)
\(\,\,\,\,\,x=2\)
\(\text{The solution is }(2,0,4).\)
\(\textbf{3)}\) \(x+y=6\)
\(\hspace{12pt}y+z=10\)
\(\hspace{12pt}x+z=8\)
\(\hspace{33pt}\)The answer is \((2,4,6)\) or \(x=2,y=4,z=6\).\(\text{Add }x+y=6\text{ and }x+z=8.\)
\(\,\,\,\,\,x+y=6\)
\(\,\,\,\,\,x+z=8\)
\(\,\,\,\,\,2x+y+z=14\)
\(\text{Because }y+z=10\text{, substitute }10\text{ for }y+z.\)
\(\,\,\,\,\,2x+10=14\)
\(\,\,\,\,\,2x=4\)
\(\,\,\,\,\,x=2\)
\(\text{Substitute }x=2\text{ into }x+y=6.\)
\(\,\,\,\,\,2+y=6\)
\(\,\,\,\,\,y=4\)
\(\text{Substitute }y=4\text{ into }y+z=10.\)
\(\,\,\,\,\,4+z=10\)
\(\,\,\,\,\,z=6\)
\(\text{The solution is }(2,4,6).\)
\(\,\,\,\,\,x+y=6\)
\(\,\,\,\,\,x+z=8\)
\(\,\,\,\,\,2x+y+z=14\)
\(\text{Because }y+z=10\text{, substitute }10\text{ for }y+z.\)
\(\,\,\,\,\,2x+10=14\)
\(\,\,\,\,\,2x=4\)
\(\,\,\,\,\,x=2\)
\(\text{Substitute }x=2\text{ into }x+y=6.\)
\(\,\,\,\,\,2+y=6\)
\(\,\,\,\,\,y=4\)
\(\text{Substitute }y=4\text{ into }y+z=10.\)
\(\,\,\,\,\,4+z=10\)
\(\,\,\,\,\,z=6\)
\(\text{The solution is }(2,4,6).\)
\(\textbf{4)}\) \(10x+y+10z=42\)
\(\hspace{12pt}11x+20y+4z=84\)
\(\hspace{12pt}x+40y+z=84\)
\(\hspace{33pt}\)The answer is \((4,2,0)\) or \(x=4,y=2,z=0\).\(\text{Solve }x+40y+z=84\text{ for }x.\)
\(\,\,\,\,\,x=84-40y-z\)
\(\text{Substitute this expression into }10x+y+10z=42.\)
\(\,\,\,\,\,10(84-40y-z)+y+10z=42\)
\(\,\,\,\,\,840-400y-10z+y+10z=42\)
\(\,\,\,\,\,840-399y=42\)
\(\,\,\,\,\,-399y=-798\)
\(\,\,\,\,\,y=2\)
\(\text{Substitute }x=84-40y-z\text{ into }11x+20y+4z=84.\)
\(\,\,\,\,\,11(84-40y-z)+20y+4z=84\)
\(\,\,\,\,\,924-440y-11z+20y+4z=84\)
\(\,\,\,\,\,-420y-7z=-840\)
\(\,\,\,\,\,60y+z=120\)
\(\text{Substitute }y=2.\)
\(\,\,\,\,\,60(2)+z=120\)
\(\,\,\,\,\,z=0\)
\(\,\,\,\,\,x+40(2)+0=84\)
\(\,\,\,\,\,x=4\)
\(\text{The solution is }(4,2,0).\)
\(\,\,\,\,\,x=84-40y-z\)
\(\text{Substitute this expression into }10x+y+10z=42.\)
\(\,\,\,\,\,10(84-40y-z)+y+10z=42\)
\(\,\,\,\,\,840-400y-10z+y+10z=42\)
\(\,\,\,\,\,840-399y=42\)
\(\,\,\,\,\,-399y=-798\)
\(\,\,\,\,\,y=2\)
\(\text{Substitute }x=84-40y-z\text{ into }11x+20y+4z=84.\)
\(\,\,\,\,\,11(84-40y-z)+20y+4z=84\)
\(\,\,\,\,\,924-440y-11z+20y+4z=84\)
\(\,\,\,\,\,-420y-7z=-840\)
\(\,\,\,\,\,60y+z=120\)
\(\text{Substitute }y=2.\)
\(\,\,\,\,\,60(2)+z=120\)
\(\,\,\,\,\,z=0\)
\(\,\,\,\,\,x+40(2)+0=84\)
\(\,\,\,\,\,x=4\)
\(\text{The solution is }(4,2,0).\)
\(\textbf{5)}\) \(3x+y-2z=10\)
\(\hspace{12pt}x+y+z=9\)
\(\hspace{12pt}x+3y+z=11\)
\(\hspace{33pt}\)The answer is \((5,1,3)\) or \(x=5,y=1,z=3\).\(\text{Multiply }x+y+z=9\text{ by }-1.\)
\(\,\,\,\,\,-x-y-z=-9\)
\(\text{Add this to }x+3y+z=11.\)
\(\,\,\,\,\,x+3y+z=11\)
\(\,\,\,\,\,-x-y-z=-9\)
\(\,\,\,\,\,2y=2\)
\(\,\,\,\,\,y=1\)
\(\text{Multiply }x+y+z=9\text{ by }-3.\)
\(\,\,\,\,\,-3x-3y-3z=-27\)
\(\text{Add this to }3x+y-2z=10.\)
\(\,\,\,\,\,3x+y-2z=10\)
\(\,\,\,\,\,-3x-3y-3z=-27\)
\(\,\,\,\,\,-2y-5z=-17\)
\(\text{Substitute }y=1.\)
\(\,\,\,\,\,-2(1)-5z=-17\)
\(\,\,\,\,\,-5z=-15\)
\(\,\,\,\,\,z=3\)
\(\,\,\,\,\,x+1+3=9\)
\(\,\,\,\,\,x=5\)
\(\text{The solution is }(5,1,3).\)
\(\,\,\,\,\,-x-y-z=-9\)
\(\text{Add this to }x+3y+z=11.\)
\(\,\,\,\,\,x+3y+z=11\)
\(\,\,\,\,\,-x-y-z=-9\)
\(\,\,\,\,\,2y=2\)
\(\,\,\,\,\,y=1\)
\(\text{Multiply }x+y+z=9\text{ by }-3.\)
\(\,\,\,\,\,-3x-3y-3z=-27\)
\(\text{Add this to }3x+y-2z=10.\)
\(\,\,\,\,\,3x+y-2z=10\)
\(\,\,\,\,\,-3x-3y-3z=-27\)
\(\,\,\,\,\,-2y-5z=-17\)
\(\text{Substitute }y=1.\)
\(\,\,\,\,\,-2(1)-5z=-17\)
\(\,\,\,\,\,-5z=-15\)
\(\,\,\,\,\,z=3\)
\(\,\,\,\,\,x+1+3=9\)
\(\,\,\,\,\,x=5\)
\(\text{The solution is }(5,1,3).\)
\(\textbf{6)}\) \(x+y=0\)
\(\hspace{12pt}y+z=-4\)
\(\hspace{12pt}x+z=6\)
\(\hspace{33pt}\)The answer is \((5,-5,1)\) or \(x=5,y=-5,z=1\).\(\text{Solve }x+y=0\text{ for }x.\)
\(\,\,\,\,\,x=-y\)
\(\text{Solve }y+z=-4\text{ for }z.\)
\(\,\,\,\,\,z=-4-y\)
\(\text{Substitute both expressions into }x+z=6.\)
\(\,\,\,\,\,-y+(-4-y)=6\)
\(\,\,\,\,\,-2y-4=6\)
\(\,\,\,\,\,-2y=10\)
\(\,\,\,\,\,y=-5\)
\(\,\,\,\,\,x=-(-5)=5\)
\(\,\,\,\,\,z=-4-(-5)=1\)
\(\text{The solution is }(5,-5,1).\)
\(\,\,\,\,\,x=-y\)
\(\text{Solve }y+z=-4\text{ for }z.\)
\(\,\,\,\,\,z=-4-y\)
\(\text{Substitute both expressions into }x+z=6.\)
\(\,\,\,\,\,-y+(-4-y)=6\)
\(\,\,\,\,\,-2y-4=6\)
\(\,\,\,\,\,-2y=10\)
\(\,\,\,\,\,y=-5\)
\(\,\,\,\,\,x=-(-5)=5\)
\(\,\,\,\,\,z=-4-(-5)=1\)
\(\text{The solution is }(5,-5,1).\)
\(\textbf{7)}\) \(x-2y+3z=4\)
\(\hspace{12pt}2x+y-4z=3\)
\(\hspace{12pt}-3x+4y-z=-2\)
\(\hspace{33pt}\)The answer is \((4,3,2)\) or \(x=4,y=3,z=2\).\(\text{Multiply }x-2y+3z=4\text{ by }-2.\)
\(\,\,\,\,\,-2x+4y-6z=-8\)
\(\text{Add this to }2x+y-4z=3.\)
\(\,\,\,\,\,2x+y-4z=3\)
\(\,\,\,\,\,-2x+4y-6z=-8\)
\(\,\,\,\,\,5y-10z=-5\)
\(\,\,\,\,\,y-2z=-1\)
\(\text{Multiply }x-2y+3z=4\text{ by }3.\)
\(\,\,\,\,\,3x-6y+9z=12\)
\(\text{Add this to }-3x+4y-z=-2.\)
\(\,\,\,\,\,-3x+4y-z=-2\)
\(\,\,\,\,\,3x-6y+9z=12\)
\(\,\,\,\,\,-2y+8z=10\)
\(\,\,\,\,\,-y+4z=5\)
\(\text{Add }y-2z=-1\text{ and }-y+4z=5.\)
\(\,\,\,\,\,2z=4\)
\(\,\,\,\,\,z=2\)
\(\,\,\,\,\,y-2(2)=-1\)
\(\,\,\,\,\,y=3\)
\(\,\,\,\,\,x-2(3)+3(2)=4\)
\(\,\,\,\,\,x=4\)
\(\text{The solution is }(4,3,2).\)
\(\,\,\,\,\,-2x+4y-6z=-8\)
\(\text{Add this to }2x+y-4z=3.\)
\(\,\,\,\,\,2x+y-4z=3\)
\(\,\,\,\,\,-2x+4y-6z=-8\)
\(\,\,\,\,\,5y-10z=-5\)
\(\,\,\,\,\,y-2z=-1\)
\(\text{Multiply }x-2y+3z=4\text{ by }3.\)
\(\,\,\,\,\,3x-6y+9z=12\)
\(\text{Add this to }-3x+4y-z=-2.\)
\(\,\,\,\,\,-3x+4y-z=-2\)
\(\,\,\,\,\,3x-6y+9z=12\)
\(\,\,\,\,\,-2y+8z=10\)
\(\,\,\,\,\,-y+4z=5\)
\(\text{Add }y-2z=-1\text{ and }-y+4z=5.\)
\(\,\,\,\,\,2z=4\)
\(\,\,\,\,\,z=2\)
\(\,\,\,\,\,y-2(2)=-1\)
\(\,\,\,\,\,y=3\)
\(\,\,\,\,\,x-2(3)+3(2)=4\)
\(\,\,\,\,\,x=4\)
\(\text{The solution is }(4,3,2).\)
\(\textbf{8)}\) \(2x+3y+z=11\)
\(\hspace{12pt}x+4y+2z=7\)
\(\hspace{12pt}3x+2y+5z=20\)
\(\hspace{33pt}\)The answer is \((5,0,1)\) or \(x=5,y=0,z=1\).\(\text{Multiply }x+4y+2z=7\text{ by }2.\)
\(\,\,\,\,\,2x+8y+4z=14\)
\(\text{Multiply }2x+3y+z=11\text{ by }-1.\)
\(\,\,\,\,\,-2x-3y-z=-11\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,5y+3z=3\)
\(\text{Multiply }x+4y+2z=7\text{ by }-3.\)
\(\,\,\,\,\,-3x-12y-6z=-21\)
\(\text{Add this to }3x+2y+5z=20.\)
\(\,\,\,\,\,3x+2y+5z=20\)
\(\,\,\,\,\,-3x-12y-6z=-21\)
\(\,\,\,\,\,-10y-z=-1\)
\(\,\,\,\,\,10y+z=1\)
\(\,\,\,\,\,z=1-10y\)
\(\text{Substitute into }5y+3z=3.\)
\(\,\,\,\,\,5y+3(1-10y)=3\)
\(\,\,\,\,\,-25y=0\)
\(\,\,\,\,\,y=0\)
\(\,\,\,\,\,z=1\)
\(\,\,\,\,\,x+4(0)+2(1)=7\)
\(\,\,\,\,\,x=5\)
\(\text{The solution is }(5,0,1).\)
\(\,\,\,\,\,2x+8y+4z=14\)
\(\text{Multiply }2x+3y+z=11\text{ by }-1.\)
\(\,\,\,\,\,-2x-3y-z=-11\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,5y+3z=3\)
\(\text{Multiply }x+4y+2z=7\text{ by }-3.\)
\(\,\,\,\,\,-3x-12y-6z=-21\)
\(\text{Add this to }3x+2y+5z=20.\)
\(\,\,\,\,\,3x+2y+5z=20\)
\(\,\,\,\,\,-3x-12y-6z=-21\)
\(\,\,\,\,\,-10y-z=-1\)
\(\,\,\,\,\,10y+z=1\)
\(\,\,\,\,\,z=1-10y\)
\(\text{Substitute into }5y+3z=3.\)
\(\,\,\,\,\,5y+3(1-10y)=3\)
\(\,\,\,\,\,-25y=0\)
\(\,\,\,\,\,y=0\)
\(\,\,\,\,\,z=1\)
\(\,\,\,\,\,x+4(0)+2(1)=7\)
\(\,\,\,\,\,x=5\)
\(\text{The solution is }(5,0,1).\)
\(\textbf{9)}\) The sum of three numbers is \(19\). The first number is double the second number. The third number is \(7\) more than the second number.
The three numbers are \(6,3,\text{ and }10\).\(\text{Let }x,y,\text{ and }z\text{ represent the first, second, and third numbers.}\)
\(\,\,\,\,\,x+y+z=19\)
\(\,\,\,\,\,x=2y\)
\(\,\,\,\,\,z=y+7\)
\(\text{Substitute }x=2y\text{ and }z=y+7\text{ into }x+y+z=19.\)
\(\,\,\,\,\,2y+y+(y+7)=19\)
\(\,\,\,\,\,4y+7=19\)
\(\,\,\,\,\,4y=12\)
\(\,\,\,\,\,y=3\)
\(\,\,\,\,\,x=2(3)=6\)
\(\,\,\,\,\,z=3+7=10\)
\(\text{The three numbers are }6,3,\text{ and }10.\)
\(\,\,\,\,\,x+y+z=19\)
\(\,\,\,\,\,x=2y\)
\(\,\,\,\,\,z=y+7\)
\(\text{Substitute }x=2y\text{ and }z=y+7\text{ into }x+y+z=19.\)
\(\,\,\,\,\,2y+y+(y+7)=19\)
\(\,\,\,\,\,4y+7=19\)
\(\,\,\,\,\,4y=12\)
\(\,\,\,\,\,y=3\)
\(\,\,\,\,\,x=2(3)=6\)
\(\,\,\,\,\,z=3+7=10\)
\(\text{The three numbers are }6,3,\text{ and }10.\)
\(\textbf{10)}\) The sum of three numbers is \(10\). The first number is \(14\) less than the second. The second number is \(3\) less than the third number.
The three numbers are \(-7,7,\text{ and }10\).\(\text{Let }x,y,\text{ and }z\text{ represent the first, second, and third numbers.}\)
\(\,\,\,\,\,x+y+z=10\)
\(\,\,\,\,\,x=y-14\)
\(\,\,\,\,\,z=y+3\)
\(\text{Substitute }x=y-14\text{ and }z=y+3\text{ into }x+y+z=10.\)
\(\,\,\,\,\,(y-14)+y+(y+3)=10\)
\(\,\,\,\,\,3y-11=10\)
\(\,\,\,\,\,3y=21\)
\(\,\,\,\,\,y=7\)
\(\,\,\,\,\,x=7-14=-7\)
\(\,\,\,\,\,z=7+3=10\)
\(\text{The three numbers are }-7,7,\text{ and }10.\)
\(\,\,\,\,\,x+y+z=10\)
\(\,\,\,\,\,x=y-14\)
\(\,\,\,\,\,z=y+3\)
\(\text{Substitute }x=y-14\text{ and }z=y+3\text{ into }x+y+z=10.\)
\(\,\,\,\,\,(y-14)+y+(y+3)=10\)
\(\,\,\,\,\,3y-11=10\)
\(\,\,\,\,\,3y=21\)
\(\,\,\,\,\,y=7\)
\(\,\,\,\,\,x=7-14=-7\)
\(\,\,\,\,\,z=7+3=10\)
\(\text{The three numbers are }-7,7,\text{ and }10.\)
\(\textbf{11)}\) \(x+y+z=4\)
\(\hspace{12pt}2x-y+z=8\)
\(\hspace{12pt}3x+2y-z=1\)
The answer is \((2,-1,3)\).
\(\text{Multiply }x+y+z=4\text{ by }-1.\)
\(\,\,\,\,\,-x-y-z=-4\)
\(\text{Add this to }2x-y+z=8.\)
\(\,\,\,\,\,2x-y+z=8\)
\(\,\,\,\,\,-x-y-z=-4\)
\(\,\,\,\,\,x-2y=4\)
\(\text{Multiply }x+y+z=4\text{ by }-3.\)
\(\,\,\,\,\,-3x-3y-3z=-12\)
\(\text{Add this to }3x+2y-z=1.\)
\(\,\,\,\,\,3x+2y-z=1\)
\(\,\,\,\,\,-3x-3y-3z=-12\)
\(\,\,\,\,\,-y-4z=-11\)
\(\,\,\,\,\,y+4z=11\)
\(\text{Solve }x-2y=4\text{ for }x.\)
\(\,\,\,\,\,x=4+2y\)
\(\text{Substitute this into }x+y+z=4.\)
\(\,\,\,\,\,4+2y+y+z=4\)
\(\,\,\,\,\,3y+z=0\)
\(\text{Multiply }3y+z=0\text{ by }-4.\)
\(\,\,\,\,\,-12y-4z=0\)
\(\text{Add this to }y+4z=11.\)
\(\,\,\,\,\,-11y=11\)
\(\,\,\,\,\,y=-1\)
\(\,\,\,\,\,z=3\)
\(\,\,\,\,\,x=2\)
\(\text{The solution is }(2,-1,3).\)
\(\,\,\,\,\,-x-y-z=-4\)
\(\text{Add this to }2x-y+z=8.\)
\(\,\,\,\,\,2x-y+z=8\)
\(\,\,\,\,\,-x-y-z=-4\)
\(\,\,\,\,\,x-2y=4\)
\(\text{Multiply }x+y+z=4\text{ by }-3.\)
\(\,\,\,\,\,-3x-3y-3z=-12\)
\(\text{Add this to }3x+2y-z=1.\)
\(\,\,\,\,\,3x+2y-z=1\)
\(\,\,\,\,\,-3x-3y-3z=-12\)
\(\,\,\,\,\,-y-4z=-11\)
\(\,\,\,\,\,y+4z=11\)
\(\text{Solve }x-2y=4\text{ for }x.\)
\(\,\,\,\,\,x=4+2y\)
\(\text{Substitute this into }x+y+z=4.\)
\(\,\,\,\,\,4+2y+y+z=4\)
\(\,\,\,\,\,3y+z=0\)
\(\text{Multiply }3y+z=0\text{ by }-4.\)
\(\,\,\,\,\,-12y-4z=0\)
\(\text{Add this to }y+4z=11.\)
\(\,\,\,\,\,-11y=11\)
\(\,\,\,\,\,y=-1\)
\(\,\,\,\,\,z=3\)
\(\,\,\,\,\,x=2\)
\(\text{The solution is }(2,-1,3).\)
\(\textbf{12)}\) \(x+2y-z=5\)
\(\hspace{12pt}3x-y+2z=-8\)
\(\hspace{12pt}2x+3y+z=9\)
The answer is \((-2,4,1)\).\(\text{Multiply }x+2y-z=5\text{ by }-3.\)
\(\,\,\,\,\,-3x-6y+3z=-15\)
\(\text{Add this to }3x-y+2z=-8.\)
\(\,\,\,\,\,3x-y+2z=-8\)
\(\,\,\,\,\,-3x-6y+3z=-15\)
\(\,\,\,\,\,-7y+5z=-23\)
\(\text{Multiply }x+2y-z=5\text{ by }-2.\)
\(\,\,\,\,\,-2x-4y+2z=-10\)
\(\text{Add this to }2x+3y+z=9.\)
\(\,\,\,\,\,2x+3y+z=9\)
\(\,\,\,\,\,-2x-4y+2z=-10\)
\(\,\,\,\,\,-y+3z=-1\)
\(\text{Multiply }-y+3z=-1\text{ by }-7.\)
\(\,\,\,\,\,7y-21z=7\)
\(\text{Add this to }-7y+5z=-23.\)
\(\,\,\,\,\,-16z=-16\)
\(\,\,\,\,\,z=1\)
\(\,\,\,\,\,-y+3(1)=-1\)
\(\,\,\,\,\,y=4\)
\(\,\,\,\,\,x+2(4)-1=5\)
\(\,\,\,\,\,x=-2\)
\(\text{The solution is }(-2,4,1).\)
\(\,\,\,\,\,-3x-6y+3z=-15\)
\(\text{Add this to }3x-y+2z=-8.\)
\(\,\,\,\,\,3x-y+2z=-8\)
\(\,\,\,\,\,-3x-6y+3z=-15\)
\(\,\,\,\,\,-7y+5z=-23\)
\(\text{Multiply }x+2y-z=5\text{ by }-2.\)
\(\,\,\,\,\,-2x-4y+2z=-10\)
\(\text{Add this to }2x+3y+z=9.\)
\(\,\,\,\,\,2x+3y+z=9\)
\(\,\,\,\,\,-2x-4y+2z=-10\)
\(\,\,\,\,\,-y+3z=-1\)
\(\text{Multiply }-y+3z=-1\text{ by }-7.\)
\(\,\,\,\,\,7y-21z=7\)
\(\text{Add this to }-7y+5z=-23.\)
\(\,\,\,\,\,-16z=-16\)
\(\,\,\,\,\,z=1\)
\(\,\,\,\,\,-y+3(1)=-1\)
\(\,\,\,\,\,y=4\)
\(\,\,\,\,\,x+2(4)-1=5\)
\(\,\,\,\,\,x=-2\)
\(\text{The solution is }(-2,4,1).\)
\(\textbf{13)}\) \(2x+y+z=9\)
\(\hspace{12pt}x-3y+2z=19\)
\(\hspace{12pt}4x+y-2z=0\)
The answer is \((3,-2,5)\).\(\text{Multiply }x-3y+2z=19\text{ by }-2.\)
\(\,\,\,\,\,-2x+6y-4z=-38\)
\(\text{Add this to }2x+y+z=9.\)
\(\,\,\,\,\,2x+y+z=9\)
\(\,\,\,\,\,-2x+6y-4z=-38\)
\(\,\,\,\,\,7y-3z=-29\)
\(\text{Multiply }x-3y+2z=19\text{ by }-4.\)
\(\,\,\,\,\,-4x+12y-8z=-76\)
\(\text{Add this to }4x+y-2z=0.\)
\(\,\,\,\,\,4x+y-2z=0\)
\(\,\,\,\,\,-4x+12y-8z=-76\)
\(\,\,\,\,\,13y-10z=-76\)
\(\text{Multiply }7y-3z=-29\text{ by }13.\)
\(\,\,\,\,\,91y-39z=-377\)
\(\text{Multiply }13y-10z=-76\text{ by }-7.\)
\(\,\,\,\,\,-91y+70z=532\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,31z=155\)
\(\,\,\,\,\,z=5\)
\(\,\,\,\,\,7y-3(5)=-29\)
\(\,\,\,\,\,y=-2\)
\(\,\,\,\,\,x-3(-2)+2(5)=19\)
\(\,\,\,\,\,x=3\)
\(\text{The solution is }(3,-2,5).\)
\(\,\,\,\,\,-2x+6y-4z=-38\)
\(\text{Add this to }2x+y+z=9.\)
\(\,\,\,\,\,2x+y+z=9\)
\(\,\,\,\,\,-2x+6y-4z=-38\)
\(\,\,\,\,\,7y-3z=-29\)
\(\text{Multiply }x-3y+2z=19\text{ by }-4.\)
\(\,\,\,\,\,-4x+12y-8z=-76\)
\(\text{Add this to }4x+y-2z=0.\)
\(\,\,\,\,\,4x+y-2z=0\)
\(\,\,\,\,\,-4x+12y-8z=-76\)
\(\,\,\,\,\,13y-10z=-76\)
\(\text{Multiply }7y-3z=-29\text{ by }13.\)
\(\,\,\,\,\,91y-39z=-377\)
\(\text{Multiply }13y-10z=-76\text{ by }-7.\)
\(\,\,\,\,\,-91y+70z=532\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,31z=155\)
\(\,\,\,\,\,z=5\)
\(\,\,\,\,\,7y-3(5)=-29\)
\(\,\,\,\,\,y=-2\)
\(\,\,\,\,\,x-3(-2)+2(5)=19\)
\(\,\,\,\,\,x=3\)
\(\text{The solution is }(3,-2,5).\)
\(\textbf{14)}\) \(x-y+2z=8\)
\(\hspace{12pt}2x+3y-z=-9\)
\(\hspace{12pt}-3x+2y+4z=-1\)
The answer is \((1,-3,2)\).\(\text{Multiply }x-y+2z=8\text{ by }-2.\)
\(\,\,\,\,\,-2x+2y-4z=-16\)
\(\text{Add this to }2x+3y-z=-9.\)
\(\,\,\,\,\,2x+3y-z=-9\)
\(\,\,\,\,\,-2x+2y-4z=-16\)
\(\,\,\,\,\,5y-5z=-25\)
\(\,\,\,\,\,y-z=-5\)
\(\text{Multiply }x-y+2z=8\text{ by }3.\)
\(\,\,\,\,\,3x-3y+6z=24\)
\(\text{Add this to }-3x+2y+4z=-1.\)
\(\,\,\,\,\,-3x+2y+4z=-1\)
\(\,\,\,\,\,3x-3y+6z=24\)
\(\,\,\,\,\,-y+10z=23\)
\(\,\,\,\,\,y=z-5\)
\(\text{Substitute into }-y+10z=23.\)
\(\,\,\,\,\,-(z-5)+10z=23\)
\(\,\,\,\,\,9z=18\)
\(\,\,\,\,\,z=2\)
\(\,\,\,\,\,y=-3\)
\(\,\,\,\,\,x-(-3)+2(2)=8\)
\(\,\,\,\,\,x=1\)
\(\text{The solution is }(1,-3,2).\)
\(\,\,\,\,\,-2x+2y-4z=-16\)
\(\text{Add this to }2x+3y-z=-9.\)
\(\,\,\,\,\,2x+3y-z=-9\)
\(\,\,\,\,\,-2x+2y-4z=-16\)
\(\,\,\,\,\,5y-5z=-25\)
\(\,\,\,\,\,y-z=-5\)
\(\text{Multiply }x-y+2z=8\text{ by }3.\)
\(\,\,\,\,\,3x-3y+6z=24\)
\(\text{Add this to }-3x+2y+4z=-1.\)
\(\,\,\,\,\,-3x+2y+4z=-1\)
\(\,\,\,\,\,3x-3y+6z=24\)
\(\,\,\,\,\,-y+10z=23\)
\(\,\,\,\,\,y=z-5\)
\(\text{Substitute into }-y+10z=23.\)
\(\,\,\,\,\,-(z-5)+10z=23\)
\(\,\,\,\,\,9z=18\)
\(\,\,\,\,\,z=2\)
\(\,\,\,\,\,y=-3\)
\(\,\,\,\,\,x-(-3)+2(2)=8\)
\(\,\,\,\,\,x=1\)
\(\text{The solution is }(1,-3,2).\)
\(\textbf{15)}\) \(x+2y-z=8\)
\(\hspace{12pt}3x-y+z=-6\)
\(\hspace{12pt}2x+3y+2z=-2\)
The answer is \((0,2,-4)\).\(\text{Multiply }x+2y-z=8\text{ by }-3.\)
\(\,\,\,\,\,-3x-6y+3z=-24\)
\(\text{Add this to }3x-y+z=-6.\)
\(\,\,\,\,\,3x-y+z=-6\)
\(\,\,\,\,\,-3x-6y+3z=-24\)
\(\,\,\,\,\,-7y+4z=-30\)
\(\text{Multiply }x+2y-z=8\text{ by }-2.\)
\(\,\,\,\,\,-2x-4y+2z=-16\)
\(\text{Add this to }2x+3y+2z=-2.\)
\(\,\,\,\,\,2x+3y+2z=-2\)
\(\,\,\,\,\,-2x-4y+2z=-16\)
\(\,\,\,\,\,-y+4z=-18\)
\(\text{Subtract }-y+4z=-18\text{ from }-7y+4z=-30.\)
\(\,\,\,\,\,-6y=-12\)
\(\,\,\,\,\,y=2\)
\(\,\,\,\,\,-2+4z=-18\)
\(\,\,\,\,\,z=-4\)
\(\,\,\,\,\,x+2(2)-(-4)=8\)
\(\,\,\,\,\,x=0\)
\(\text{The solution is }(0,2,-4).\)
\(\,\,\,\,\,-3x-6y+3z=-24\)
\(\text{Add this to }3x-y+z=-6.\)
\(\,\,\,\,\,3x-y+z=-6\)
\(\,\,\,\,\,-3x-6y+3z=-24\)
\(\,\,\,\,\,-7y+4z=-30\)
\(\text{Multiply }x+2y-z=8\text{ by }-2.\)
\(\,\,\,\,\,-2x-4y+2z=-16\)
\(\text{Add this to }2x+3y+2z=-2.\)
\(\,\,\,\,\,2x+3y+2z=-2\)
\(\,\,\,\,\,-2x-4y+2z=-16\)
\(\,\,\,\,\,-y+4z=-18\)
\(\text{Subtract }-y+4z=-18\text{ from }-7y+4z=-30.\)
\(\,\,\,\,\,-6y=-12\)
\(\,\,\,\,\,y=2\)
\(\,\,\,\,\,-2+4z=-18\)
\(\,\,\,\,\,z=-4\)
\(\,\,\,\,\,x+2(2)-(-4)=8\)
\(\,\,\,\,\,x=0\)
\(\text{The solution is }(0,2,-4).\)
\(\textbf{16)}\) \(2x-y+z=-3\)
\(\hspace{12pt}x+2y-3z=-1\)
\(\hspace{12pt}3x+y+2z=4\)
The answer is \((-1,3,2)\).\(\text{Multiply }x+2y-3z=-1\text{ by }2.\)
\(\,\,\,\,\,2x+4y-6z=-2\)
\(\text{Multiply }2x-y+z=-3\text{ by }-1.\)
\(\,\,\,\,\,-2x+y-z=3\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,5y-7z=1\)
\(\text{Multiply }3x+y+2z=4\text{ by }2.\)
\(\,\,\,\,\,6x+2y+4z=8\)
\(\text{Multiply }2x-y+z=-3\text{ by }-3.\)
\(\,\,\,\,\,-6x+3y-3z=9\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,5y+z=17\)
\(\text{Subtract }5y-7z=1\text{ from }5y+z=17.\)
\(\,\,\,\,\,8z=16\)
\(\,\,\,\,\,z=2\)
\(\,\,\,\,\,5y+2=17\)
\(\,\,\,\,\,y=3\)
\(\,\,\,\,\,2x-3+2=-3\)
\(\,\,\,\,\,x=-1\)
\(\text{The solution is }(-1,3,2).\)
\(\,\,\,\,\,2x+4y-6z=-2\)
\(\text{Multiply }2x-y+z=-3\text{ by }-1.\)
\(\,\,\,\,\,-2x+y-z=3\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,5y-7z=1\)
\(\text{Multiply }3x+y+2z=4\text{ by }2.\)
\(\,\,\,\,\,6x+2y+4z=8\)
\(\text{Multiply }2x-y+z=-3\text{ by }-3.\)
\(\,\,\,\,\,-6x+3y-3z=9\)
\(\text{Add the equations.}\)
\(\,\,\,\,\,5y+z=17\)
\(\text{Subtract }5y-7z=1\text{ from }5y+z=17.\)
\(\,\,\,\,\,8z=16\)
\(\,\,\,\,\,z=2\)
\(\,\,\,\,\,5y+2=17\)
\(\,\,\,\,\,y=3\)
\(\,\,\,\,\,2x-3+2=-3\)
\(\,\,\,\,\,x=-1\)
\(\text{The solution is }(-1,3,2).\)
\(\textbf{17)}\) A theater sold \(120\) adult, student, and senior tickets. It sold \(20\) more adult tickets than senior tickets. Adult tickets cost \(\$12\), student tickets cost \(\$8\), and senior tickets cost \(\$6\). The theater collected \(\$1{,}100\). How many of each ticket were sold?
The theater sold \(50\) adult, \(40\) student, and \(30\) senior tickets.\(\text{Let }a,s,\text{ and }r\text{ represent adult, student, and senior tickets.}\)
\(\,\,\,\,\,a+s+r=120\)
\(\,\,\,\,\,a-r=20\)
\(\,\,\,\,\,12a+8s+6r=1100\)
\(\text{Solve }a-r=20\text{ for }a.\)
\(\,\,\,\,\,a=r+20\)
\(\text{Substitute into }a+s+r=120.\)
\(\,\,\,\,\,(r+20)+s+r=120\)
\(\,\,\,\,\,s=100-2r\)
\(\text{Substitute both expressions into the revenue equation.}\)
\(\,\,\,\,\,12(r+20)+8(100-2r)+6r=1100\)
\(\,\,\,\,\,12r+240+800-16r+6r=1100\)
\(\,\,\,\,\,2r+1040=1100\)
\(\,\,\,\,\,r=30\)
\(\,\,\,\,\,a=30+20=50\)
\(\,\,\,\,\,s=100-2(30)=40\)
\(\text{The theater sold }50\text{ adult, }40\text{ student, and }30\text{ senior tickets.}\)
\(\,\,\,\,\,a+s+r=120\)
\(\,\,\,\,\,a-r=20\)
\(\,\,\,\,\,12a+8s+6r=1100\)
\(\text{Solve }a-r=20\text{ for }a.\)
\(\,\,\,\,\,a=r+20\)
\(\text{Substitute into }a+s+r=120.\)
\(\,\,\,\,\,(r+20)+s+r=120\)
\(\,\,\,\,\,s=100-2r\)
\(\text{Substitute both expressions into the revenue equation.}\)
\(\,\,\,\,\,12(r+20)+8(100-2r)+6r=1100\)
\(\,\,\,\,\,12r+240+800-16r+6r=1100\)
\(\,\,\,\,\,2r+1040=1100\)
\(\,\,\,\,\,r=30\)
\(\,\,\,\,\,a=30+20=50\)
\(\,\,\,\,\,s=100-2(30)=40\)
\(\text{The theater sold }50\text{ adult, }40\text{ student, and }30\text{ senior tickets.}\)
\(\textbf{18)}\) A jar contains \(40\) quarters, dimes, and nickels worth a total of \(\$5.00\). There are twice as many dimes as nickels. How many of each coin are in the jar?
There are \(10\) quarters, \(20\) dimes, and \(10\) nickels.\(\text{Let }q,d,\text{ and }n\text{ represent quarters, dimes, and nickels.}\)
\(\,\,\,\,\,q+d+n=40\)
\(\,\,\,\,\,25q+10d+5n=500\)
\(\,\,\,\,\,d=2n\)
\(\text{Substitute }d=2n\text{ into }q+d+n=40.\)
\(\,\,\,\,\,q+2n+n=40\)
\(\,\,\,\,\,q=40-3n\)
\(\text{Substitute }q=40-3n\text{ and }d=2n\text{ into the value equation.}\)
\(\,\,\,\,\,25(40-3n)+10(2n)+5n=500\)
\(\,\,\,\,\,1000-75n+20n+5n=500\)
\(\,\,\,\,\,1000-50n=500\)
\(\,\,\,\,\,-50n=-500\)
\(\,\,\,\,\,n=10\)
\(\,\,\,\,\,d=2(10)=20\)
\(\,\,\,\,\,q=40-3(10)=10\)
\(\text{There are }10\text{ quarters, }20\text{ dimes, and }10\text{ nickels.}\)
\(\,\,\,\,\,q+d+n=40\)
\(\,\,\,\,\,25q+10d+5n=500\)
\(\,\,\,\,\,d=2n\)
\(\text{Substitute }d=2n\text{ into }q+d+n=40.\)
\(\,\,\,\,\,q+2n+n=40\)
\(\,\,\,\,\,q=40-3n\)
\(\text{Substitute }q=40-3n\text{ and }d=2n\text{ into the value equation.}\)
\(\,\,\,\,\,25(40-3n)+10(2n)+5n=500\)
\(\,\,\,\,\,1000-75n+20n+5n=500\)
\(\,\,\,\,\,1000-50n=500\)
\(\,\,\,\,\,-50n=-500\)
\(\,\,\,\,\,n=10\)
\(\,\,\,\,\,d=2(10)=20\)
\(\,\,\,\,\,q=40-3(10)=10\)
\(\text{There are }10\text{ quarters, }20\text{ dimes, and }10\text{ nickels.}\)
Challenge Problems
\(\textbf{19)}\) Determine whether the system has one solution, no solution, or infinitely many solutions.
\(\hspace{12pt}x+y+z=6\)
\(\hspace{12pt}2x-y+3z=14\)
\(\hspace{12pt}3x+4z=21\)
The system has no solution.\(\text{Add }x+y+z=6\text{ and }2x-y+3z=14.\)
\(\,\,\,\,\,x+y+z=6\)
\(\,\,\,\,\,2x-y+3z=14\)
\(\,\,\,\,\,3x+4z=20\)
\(\text{However, the third equation is }3x+4z=21.\)
\(\text{The same expression cannot equal both }20\text{ and }21.\)
\(\text{Therefore, the system has no solution.}\)
\(\,\,\,\,\,x+y+z=6\)
\(\,\,\,\,\,2x-y+3z=14\)
\(\,\,\,\,\,3x+4z=20\)
\(\text{However, the third equation is }3x+4z=21.\)
\(\text{The same expression cannot equal both }20\text{ and }21.\)
\(\text{Therefore, the system has no solution.}\)
\(\textbf{20)}\) Determine whether the system has one solution, no solution, or infinitely many solutions.
\(\hspace{12pt}x+y+z=6\)
\(\hspace{12pt}x-y+z=2\)
\(\hspace{12pt}2x+2z=8\)
The system has infinitely many solutions: \((x,y,z)=(4-t,2,t)\).\(\text{Add }x+y+z=6\text{ and }x-y+z=2.\)
\(\,\,\,\,\,x+y+z=6\)
\(\,\,\,\,\,x-y+z=2\)
\(\,\,\,\,\,2x+2z=8\)
\(\text{This result is identical to the third equation.}\)
\(\text{Therefore, the third equation does not provide any new information.}\)
\(\text{Multiply }x-y+z=2\text{ by }-1.\)
\(\,\,\,\,\,-x+y-z=-2\)
\(\text{Add this to }x+y+z=6.\)
\(\,\,\,\,\,x+y+z=6\)
\(\,\,\,\,\,-x+y-z=-2\)
\(\,\,\,\,\,2y=4\)
\(\,\,\,\,\,y=2\)
\(\text{Let }z=t.\)
\(\,\,\,\,\,x+2+t=6\)
\(\,\,\,\,\,x=4-t\)
\(\text{The solutions are }(x,y,z)=(4-t,2,t).\)
\(\text{Therefore, the system has infinitely many solutions.}\)
\(\,\,\,\,\,x+y+z=6\)
\(\,\,\,\,\,x-y+z=2\)
\(\,\,\,\,\,2x+2z=8\)
\(\text{This result is identical to the third equation.}\)
\(\text{Therefore, the third equation does not provide any new information.}\)
\(\text{Multiply }x-y+z=2\text{ by }-1.\)
\(\,\,\,\,\,-x+y-z=-2\)
\(\text{Add this to }x+y+z=6.\)
\(\,\,\,\,\,x+y+z=6\)
\(\,\,\,\,\,-x+y-z=-2\)
\(\,\,\,\,\,2y=4\)
\(\,\,\,\,\,y=2\)
\(\text{Let }z=t.\)
\(\,\,\,\,\,x+2+t=6\)
\(\,\,\,\,\,x=4-t\)
\(\text{The solutions are }(x,y,z)=(4-t,2,t).\)
\(\text{Therefore, the system has infinitely many solutions.}\)
Related Pages\(\)
\(\bullet\text{ Algebra 2/Precalc Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ 3 Equation System Calculator}\)
\(\,\,\,\,\,\,\,\,\text{(wolframalpha.com)}\)
\(\bullet\text{ Solving Systems with Substitution}\)
\(\,\,\,\,\,\,\,\,y=-2x+5\)
\(\,\,\,\,\,\,\,\,y=x-1…\)
\(\bullet\text{ Solving Systems with Elimination}\)
\(\,\,\,\,\,\,\,\,7x+4y=31\)
\(\,\,\,\,\,\,\,\,3x+2y=15…\)
\(\bullet\text{ Graphing Systems of Inequalities}\)
\(\,\,\,\,\,\,\,\,y\lt-3x+2\)
\(\,\,\,\,\,\,\,\,y\ge\frac{1}{2}x-1…\)
\(\bullet\text{ Nonlinear Systems}\)
\(\,\,\,\,\,\,\,\,x^2+y^2=8\)
\(\,\,\,\,\,\,\,\,y=x…\)
\(\bullet\text{ Andymath Homepage}\)
