3 Variable Simultaneous Equations – Substitution Method

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Question 1 of 2

1. Question

Solve the following simultaneous equations by substitution.

`a-2b+c=-4`

`2a+2b-c=10`

`-3a-b-4c=9`

`a=` (2)

`b=` (1)

`c=` (-4)

Hint

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Substitution Method

`1)` Make one variable the subject in one equation
`2)` Substitute this variable in to the other two equations
`3)` Solve these two equations using substitution and find `2` variables
`4)` Substitute the `2` variables into the original equation

First, label the equations `1`, `2` and `3` respectively.

`a-2b+c`	`=`	`-4`	Equation `1`
`2a+2b-c`	`=`	`10`	Equation `2`
`-3a-b-4c`	`=`	`9`	Equation `3`

Next, make `a` the subject in Equation `1`.

`a-2b+c`	`=`	`-4`
`a-2b+c` `+(2b-c)`	`=`	`-4` `+(2b-c)`	Add `2b-c` to both sides
`a`	`=`	`2b-c-4`	Simplify

Substitute Equation `1`’s `a` into both Equations `2` and `3`. Label the results as Equations `A` and `B`

`2``a``+2b-c`	`=`	`10`	Equation `2`
`2(``2b-c-4``)+2b-c`	`=`	`10`	`a=2b-c-4`
`4b-2c-8+2b-c`	`=`	`10`	Distribute `2` inside the parenthesis
`6b-3c-8`	`=`	`10`	Simplify
`6b-3c-8` `+8`	`=`	`10` `+8`	Add `8` to both sides
`6b-3c`	`=`	`18`	Equation `A`

`-3``a``-b-4c`	`=`	`9`	Equation `3`
`-3(``2b-c-4``)-b-4c`	`=`	`9`	`a=2b-c-4`
`-6b+3c+12-b-4c`	`=`	`9`	Distribute `2` inside the parenthesis
`-7b-c+12`	`=`	`9`	Simplify
`-7b-c+12` `-12`	`=`	`9` `-12`	Subtract `12` from both sides
`-7b-c`	`=`	`-3`	Equation `B`

Next, make `c` the subject in Equation `B`.

`-7b-c`	`=`	`-3`	Equation `B`
`-7b-c` `+c`	`=`	`-3` `+c`	Add `c` to both sides
`-7b`	`=`	`-3+c`
`-7b` `+3`	`=`	`-3+c` `+3`	Add `3` to both sides
`-7b+3`	`=`	`c`
`c`	`=`	`-7b+3`

Now, substitute `c=-7b+3` into Equation `A` and solve for `b`.

`6b-3``c`	`=`	`18`	Equation `A`
`6b-3(``-7b+3``)`	`=`	`18`	`c=-7b+3`
`6b+21b-9`	`=`	`18`	Distribute `3` inside the parenthesis
`27b-9`	`=`	`18`	Simplify
`27b-9` `+9`	`=`	`18` `+9`	Add `9` to both sides
`27b`	`=`	`27`
`27b` `div27`	`=`	`27` `div27`	Divide both sides by `27`
`b`	`=`	`1`

Next, substitute the value of `b` into Equation `B` and solve for `c`

`c`	`=`	`-7``b` `+3`	Equation `B`
`c`	`=`	`-7(``1``)+3`	`b=1`
`c`	`=`	`-7+3`
`c`	`=`	`-4`

Finally, substitute the known values of `b` and `c` into Equation `1` to solve for `a`.

`a-2``b``+``c`	`=`	`-4`	Equation `1`
`a-2(``1``)+``-4`	`=`	`-4`	`b=1` and `c=-4`
`a-2-4`	`=`	`-4`
`a-6`	`=`	`-4`	Simplify
`a-6` `+6`	`=`	`-4` `+6`	Add `6` to both sides
`a`	`=`	`2`

`a=2`

`b =1`

`c =-4`

Question 2 of 2

2. Question

Solve the following simultaneous equations by substitution.

`3x-3y+z=-2`

`2y-z=-5`

`x+4y+2z=-9`

`x=` (-3)

`y=` (-2)

`z=` (1)

Hint

Help Video

Correct

Correct!

Incorrect

Substitution Method

`1)` Make one variable the subject in one equation
`2)` Substitute this variable in to the other two equations
`3)` Solve these two equations using substitution and find `2` variables
`4)` Substitute the `2` variables into the original equation

First, label the equations `1`, `2` and `3` respectively.

`3x-3y+z`	`=`	`-2`	Equation `1`
`2y-z`	`=`	`-5`	Equation `2`
`x+4y+2z`	`=`	`-9`	Equation `3`

Next, make `z` the subject in Equation `1`.

`2y-z`	`=`	`-5`
`2y-z` `+5`	`=`	`-5` `+5`	Add `5` to both sides
`2y-z+5`	`=`	`0`
`2y+5-z` `+z`	`=`	`0` `+z`	Add `z` to both sides
`2y+5`	`=`	`z`
`z`	`=`	`2y+5`	Simplify

Substitute `z=2y+5` into both Equations `1` and `3`. Label the results as Equations `A` and `B`

`3x-3y+``z`	`=`	`-2`	Equation `1`
`3x-3y+``2y+5`	`=`	`-2`	`z=2y+5`
`3x-y+5`	`=`	`-2`	Combine like terms
`3x-y+5` `-5`	`=`	`-2` `-5`	Subtract `5` from both sides
`3x-y`	`=`	`-7`	Equation `A`

`x+4y+2``z`	`=`	`-9`	Equation `3`
`x+4y+2(``2y+5``)`	`=`	`-9`	`z=2y+5`
`x+4y+4y+10`	`=`	`-9`	Distribute `2` inside the parenthesis
`x+8y+10`	`=`	`-9`	Combine like terms
`x+8y+10` `-10`	`=`	`-9` `-10`	Subtract `10` from both sides
`x+8y`	`=`	`-19`	Equation `B`

Next, make `y` the subject in Equation `A`.

`3x-y`	`=`	`-7`	Equation `A`
`3x-y` `+7`	`=`	`-7` `+7`	Add `7` to both sides
`3x+7-y`	`=`	`0`
`3x+7-y` `+y`	`=`	`0` `+y`	Add `y` to both sides
`3x+7`	`=`	`y`
`y`	`=`	`3x+7`

Now, substitute `y=3x+7` into Equation `B` and solve for `x`.

`x+8``y`	`=`	`-19`	Equation `B`
`x+8(``3x+7``)`	`=`	`-19`	`y=3x+7`
`x+24x+56`	`=`	`-19`	Distribute `8` inside the parenthesis
`25x+56`	`=`	`-19`	Combine like terms
`25x+56` `-56`	`=`	`-19` `-56`	Subtract `56` from both sides
`25x`	`=`	`-75`
`27x` `div25`	`=`	`-75` `div25`	Divide both sides by `25`
`x`	`=`	`-3`

Next, substitute the value of `x` into Equation `A` and solve for `y`

`3``x``-y`	`=`	`-7`	Equation `A`
`3(``-3``)-y`	`=`	`-7`	`x=-3`
`-9-y`	`=`	`-7`
`-9-y` `+9`	`=`	`-7` `+9`	Add `9` to both sides
`-y`	`=`	`2`
`-y` `times(-1)`	`=`	`2` `times(-1)`	Multiply both sides by `-1`
`y`	`=`	`-2`

Finally, substitute the known value of `y` into Equation `2` to solve for `z`.

`2``y``-z`	`=`	`-5`	Equation `2`
`2(``-2``)-z`	`=`	`-5`	`y=-2`
`-4-z`	`=`	`-5`
`-4-z` `+4`	`=`	`-5` `+4`	Add `4` to both sides
`-z`	`=`	`-1`
`-z` `times(-1)`	`=`	`-1` `times(-1)`	Multiply both sides by `-1`
`z`	`=`	`1`

`x=-3`

`y =-2`

`z =1`

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