Algebra - 3 or More Simultaneous Equations
Question 1
Solve the following
a) $\begin{align}x+y+z&=15\\x+y&=7\\x-y&=3\end{align}$
a
$x=5,y=2,z=8$
Question ID: 20240010010
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$\begin{align}x+y+z&=15\qquad(1)\\x+y&=7\qquad(2)\\x-y&=3\qquad(3)\\ \\(2)+(3)\qquad 2x&=10\\x&=5\qquad(4)\\\\(4)\text{ into } (2)\qquad5+y&=7\\y&=2\qquad(5)\\\\(4)\text{ and }(5) \text{ into }(1)\qquad5+2+z&=15\\7+z&=15\\z&=8\\\\x=5,y&=2,z=8\end{align}$
Question ID: 20240010010
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b) $\begin{align}a-b-c&=4\\a+b+c&=14\\b-c&=3\end{align}$
a
$a=9,b=4,c=1$
Question ID: 20240010020
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c) $\begin{align}2p+q+r&=33\\p+q+r&=25\\5q-r&=7\end{align}$
a
$p=8,q=4,r=13$
Question ID: 20240010030
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d) $\begin{align}5y-x+2z&=45\\2x-5z&=14\\z-5x&=11\end{align}$
a
$x=-3,y=10,z=-4$
Question ID: 20240010040
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Question 2
Solve the following
a) $\begin{align}x+2y-3z&=-10\\x-y+z&=10\\x-y+2z&=18\end{align}$
a
$x=6,y=4,z=8$
Question ID: 20240020010
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b) $\begin{align}5s-t+3u&=52\\s+t+u&=20\\4s-2t+u&=29\end{align}$
a
$s=10,t=7,u=3$
Question ID: 20240020020
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Question 3
Solve the following
a) $\begin{align}x+y+z&=11\\x+y-z&=9\\2x+y-z&=14\end{align}$
a
$x=5,y=5,z=1$
Question ID: 20240030010
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b) $\begin{align}3a+4b+2c&=39\\5a-b-2c&=6\\4a+10b+2c&=84\end{align}$
a
$a=3,b=7,c=1$
Question ID: 20240030020
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c) $\begin{align}4x-5y-8z&=-154\\10x+5y+6z&=240\\x-5y-2z&=-61\end{align}$
a
$x=9,y=6,z=20$
Question ID: 20240030030
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d) $\begin{align}4p-q-r&=50\\r+q-2p&=-20\\5q-3p-r&=-19\end{align}$
a
$p=15,q=6,r=4$
Question ID: 20240030040
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Question 4
Solve the following
a) $\begin{align}5x+2y+8z&=77\\3x+y-3z&=7\\4x-4y+z&=29\end{align}$
a
$x=7,y=1,z=5$
Question ID: 20240040010
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b) $\begin{align}2x+6y-z&=47\\6x-2y+5z&=37\\8x-4y+3z&=13\end{align}$
a
$x=3,y=8,z=7$
Question ID: 20240040020
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c) $\begin{align}p-3q-5r&=-61\\12q+3p+2r&=46\\6p-5q+8r&=78\end{align}$
a
$p=0,q=2,r=11$
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d) $\begin{align}2a-5b-10c&=-49\\3a+10b+7c&=50\\11a-b+5c&=131\end{align}$
a
$a=8,b=-3,c=8$
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Question 5
Solve the following
a) $\begin{align}x+2y-z&=-8\\3x+3y-2z&=-9\\5x-y+7z&=90\end{align}$
a
$x=5,y=-2,z=9$
Question ID: 20240050010
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b) $\begin{align}2x-3y+5z&=38\\5x+5y+9z&=67\\4x-4y-6z&=-52\end{align}$
a
$x=-1,y=0,z=8$
Question ID: 20240050020
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Question 6
Solve the following
a) $\begin{align}3a-b+2c-d&=18\\7a+b+3c+2d&=54\\2a+b+8c+6d&=57\\4a-b-3c+d&=7\end{align}$
a
$a=5,b=3,c=4,d=2$
Question ID: 20240060010
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b) $\begin{align}p-2q+4r-s&=-71\\2p-3q+10r-5s&=-161\\3p+5q+2r+s&=56\\5p-q+3r+2s&=-42\end{align}$
a
$p=-3,q=15,r=-8,s=6$
Question ID: 20240060020
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Question 1
Solve the following
a) $\begin{align}x+y+z&=15\\x+y&=7\\x-y&=3\end{align}$
b) $\begin{align}a-b-c&=4\\a+b+c&=14\\b-c&=3\end{align}$
c) $\begin{align}2p+q+r&=33\\p+q+r&=25\\5q-r&=7\end{align}$
d) $\begin{align}5y-x+2z&=45\\2x-5z&=14\\z-5x&=11\end{align}$
Question 2
Solve the following
a) $\begin{align}x+2y-3z&=-10\\x-y+z&=10\\x-y+2z&=18\end{align}$
b) $\begin{align}5s-t+3u&=52\\s+t+u&=20\\4s-2t+u&=29\end{align}$
Question 3
Solve the following
a) $\begin{align}x+y+z&=11\\x+y-z&=9\\2x+y-z&=14\end{align}$
b) $\begin{align}3a+4b+2c&=39\\5a-b-2c&=6\\4a+10b+2c&=84\end{align}$
c) $\begin{align}4x-5y-8z&=-154\\10x+5y+6z&=240\\x-5y-2z&=-61\end{align}$
d) $\begin{align}4p-q-r&=50\\r+q-2p&=-20\\5q-3p-r&=-19\end{align}$
Question 4
Solve the following
a) $\begin{align}5x+2y+8z&=77\\3x+y-3z&=7\\4x-4y+z&=29\end{align}$
b) $\begin{align}2x+6y-z&=47\\6x-2y+5z&=37\\8x-4y+3z&=13\end{align}$
c) $\begin{align}p-3q-5r&=-61\\12q+3p+2r&=46\\6p-5q+8r&=78\end{align}$
d) $\begin{align}2a-5b-10c&=-49\\3a+10b+7c&=50\\11a-b+5c&=131\end{align}$
Question 5
Solve the following
a) $\begin{align}x+2y-z&=-8\\3x+3y-2z&=-9\\5x-y+7z&=90\end{align}$
b) $\begin{align}2x-3y+5z&=38\\5x+5y+9z&=67\\4x-4y-6z&=-52\end{align}$
Question 6
Solve the following
a) $\begin{align}3a-b+2c-d&=18\\7a+b+3c+2d&=54\\2a+b+8c+6d&=57\\4a-b-3c+d&=7\end{align}$
b) $\begin{align}p-2q+4r-s&=-71\\2p-3q+10r-5s&=-161\\3p+5q+2r+s&=56\\5p-q+3r+2s&=-42\end{align}$
Answers
Question 1
a) $x=5,y=2,z=8$
b) $a=9,b=4,c=1$
c) $p=8,q=4,r=13$
d) $x=-3,y=10,z=-4$
Question 2
a) $x=6,y=4,z=8$
b) $s=10,t=7,u=3$
Question 3
a) $x=5,y=5,z=1$
b) $a=3,b=7,c=1$
c) $x=9,y=6,z=20$
d) $p=15,q=6,r=4$
Question 4
a) $x=7,y=1,z=5$
b) $x=3,y=8,z=7$
c) $p=0,q=2,r=11$
d) $a=8,b=-3,c=8$
Question 5
a) $x=5,y=-2,z=9$
b) $x=-1,y=0,z=8$
Question 6
a) $a=5,b=3,c=4,d=2$
b) $p=-3,q=15,r=-8,s=6$
Question 1
Solve the following
a) $\begin{align}x+y+z&=15\\x+y&=7\\x-y&=3\end{align}$
b) $\begin{align}a-b-c&=4\\a+b+c&=14\\b-c&=3\end{align}$
c) $\begin{align}2p+q+r&=33\\p+q+r&=25\\5q-r&=7\end{align}$
d) $\begin{align}5y-x+2z&=45\\2x-5z&=14\\z-5x&=11\end{align}$
Question 2
Solve the following
a) $\begin{align}x+2y-3z&=-10\\x-y+z&=10\\x-y+2z&=18\end{align}$
b) $\begin{align}5s-t+3u&=52\\s+t+u&=20\\4s-2t+u&=29\end{align}$
Question 3
Solve the following
a) $\begin{align}x+y+z&=11\\x+y-z&=9\\2x+y-z&=14\end{align}$
b) $\begin{align}3a+4b+2c&=39\\5a-b-2c&=6\\4a+10b+2c&=84\end{align}$
c) $\begin{align}4x-5y-8z&=-154\\10x+5y+6z&=240\\x-5y-2z&=-61\end{align}$
d) $\begin{align}4p-q-r&=50\\r+q-2p&=-20\\5q-3p-r&=-19\end{align}$
Question 4
Solve the following
a) $\begin{align}5x+2y+8z&=77\\3x+y-3z&=7\\4x-4y+z&=29\end{align}$
b) $\begin{align}2x+6y-z&=47\\6x-2y+5z&=37\\8x-4y+3z&=13\end{align}$
c) $\begin{align}p-3q-5r&=-61\\12q+3p+2r&=46\\6p-5q+8r&=78\end{align}$
d) $\begin{align}2a-5b-10c&=-49\\3a+10b+7c&=50\\11a-b+5c&=131\end{align}$
Question 5
Solve the following
a) $\begin{align}x+2y-z&=-8\\3x+3y-2z&=-9\\5x-y+7z&=90\end{align}$
b) $\begin{align}2x-3y+5z&=38\\5x+5y+9z&=67\\4x-4y-6z&=-52\end{align}$
Question 6
Solve the following
a) $\begin{align}3a-b+2c-d&=18\\7a+b+3c+2d&=54\\2a+b+8c+6d&=57\\4a-b-3c+d&=7\end{align}$
b) $\begin{align}p-2q+4r-s&=-71\\2p-3q+10r-5s&=-161\\3p+5q+2r+s&=56\\5p-q+3r+2s&=-42\end{align}$
Answers
Question 1
a) $x=5,y=2,z=8$
b) $a=9,b=4,c=1$
c) $p=8,q=4,r=13$
d) $x=-3,y=10,z=-4$
Question 2
a) $x=6,y=4,z=8$
b) $s=10,t=7,u=3$
Question 3
a) $x=5,y=5,z=1$
b) $a=3,b=7,c=1$
c) $x=9,y=6,z=20$
d) $p=15,q=6,r=4$
Question 4
a) $x=7,y=1,z=5$
b) $x=3,y=8,z=7$
c) $p=0,q=2,r=11$
d) $a=8,b=-3,c=8$
Question 5
a) $x=5,y=-2,z=9$
b) $x=-1,y=0,z=8$
Question 6
a) $a=5,b=3,c=4,d=2$
b) $p=-3,q=15,r=-8,s=6$
Solutions
Question 1
a) $\begin{align}x+y+z&=15\qquad(1)\\x+y&=7\qquad(2)\\x-y&=3\qquad(3)\\ \\(2)+(3)\qquad 2x&=10\\x&=5\qquad(4)\\\\(4)\text{ into } (2)\qquad5+y&=7\\y&=2\qquad(5)\\\\(4)\text{ and }(5) \text{ into }(1)\qquad5+2+z&=15\\7+z&=15\\z&=8\\\\x=5,y&=2,z=8\end{align}$
b)
c)
d)
Question 2
a)
b)
Question 3
a)
b)
c)
d)
Question 4
a)
b)
c)
d)
Question 5
a)
b)
Question 6
a)
b)
