These are both methods of solving a system of equations with variables, for example:

x + 2y = 7

2x + y = 8

The point of substitution and elimination is to find the values of the variables by applying various mathematical methods to the equations so that the variable and its value are the only things left. It can also be used for solving equations with three or more variables, or with three or more equations. We will first look at substitution.

The first method of substitution we will look at is solving two equations with two variables. The idea of substitution is to isolate one variable and place the corresponding value in the other equation. For example:

1. 3x - 6y = 9

2. 2x + 3y = 27

We will isolate the variable x in the first equation by moving the other variable (y) to the other side of the equation and eliminating xs coefficient (3). A trick for doing this is to glance at your equations and find which variable will divide most evenly into the other terms of the equation, giving you the simplest numbers to work with and avoiding awful things like decimals and fractions, which are terrible and horrible:

3x ** 6y** = 9 -6y
goes to the other side, becoming +6y

3x **/ 3**= (9 + 6y) **/ 3 **Both
sides are divided by 3 to eliminate the 3 in front of the x

x = 3 + 2y

Now that we have a value for x, we can place our new x value (3 + 2y) from our first equation in the place of x in our second equation and solve for y to get its value:

2**x** + 3y = 27 Substitute
(3 + 2y) where x is

**2(3 + 2y)** + 3y = 27 Multiply out the bracket to get your
new values

6 + **4y + 3y** = 27 Combine the
y terms

**6** + 7y = 27 Move your 6
to the other side of the equation, making it 6

7y = **27 6 **Subtract 6 from 27

7y **/ 7**= 21 **/ 7** Divide both sides by seven to get the
y value

y = 3

Now that you know that y=3, substitute that value into either of the first two equations and solve for x, which will solve our equations. We will use the first equation:

3x 6**y** = 9 Substitute 3 for y

3x **6(3)** = 9 Multiply 6 by 3

3x ** 18** = 9 Move 18 to
the other side of the equation, making it +18

3x = **9 + 18 **Add 9 and 18

3x **/ 3** = 27 **/ 3** Divide both sides by three to get the
value of x

x = 9

The equations have now been solved using substitution. Our answer will read:

**x = 9**

**y = 3**

** **

Its a good idea to substitute both numbers into the equations to make sure they equal out:

1. 3(9) 6(3) = 9

27 18 = 9

9 = 9

2. 2(9) + 3(3) = 27

18 + 9 = 27

27 = 27

**Examples**

Here are some examples of equations we will solve with substitution, progressing from easier to more difficult.

**Example I:** These are very simple equations to solve, as the
variables have no coefficients, cutting down possible complications:

1. x + y = 11

2. x y = 3

We must single out one variable to substitute in the other equation, and we will use the second equation and single out x:

x ** y** = 3 Move
the y to the other side, making it +y

x = 3 + y

Now substitute that value into the first equation in the place of x and solve for y:

**x** + y = 11 Substitute
(3 + y) in the place of x

**(3 + y)** + y =
11 Drop your brackets

3 **+ y + y** = 11 Combine
your y variables

**3** + 2y = 11 Move your 3 to the other side, becoming 3

2y = **11 3** Subtract 3 from 11

2y **/ 2** = 8 **/ 2** Divide both sides by 2 to get
the value of y

y = 4

Now that you have the value for y, substitute it into one of your first two equations. We will use the second:

x **y** = 3 Substitute
4 for y

x ** 4** = 3 Move 4 to the other side,
becoming +4

x = 7

**x = 7**

**y = 4**

**Example II:** These are regular equations, like the two used in the
explanation. They are the most common
type of equations you will solve:

1. 4x + 3y = 20

2. 2x + 6y = 28

We need to get one variable alone so we can substitute, so we will use x in the second equation as it will divide out the nicest:

2x **+ 6y** = 28 Move
6y to the other side, becoming 6y

2x **/ 2** = (28
6y) **/ 2** Divide both sides by 2 to get your x value

x = 14 3y

Our new x value can now be placed in the first equation and we can solve for y:

4**x** + 3y = 20 Substitute
(14 3y) in place of x

**4(14 3y)** + 3y
= 20 Multiply out your
brackets with the 4

56 **12y + 3y** =
20 Combine your y
variables

**56** 9y = 20 Move
56 to the other side, becoming -56

- 9y = **20 56** Subtract
56 from 20

- 9y **/ -9**= - 36
**/ -9** Divide both sides by 9 to
get the y value

y = 4

Now place substitute your y value in one of the first two equations and solve for x. We will use the second equation:

2x + 6**y** = 28 Substitute
4 for y

2x + **6(4) **= 28 Multiply
6 by 4

2x **+ 24** = 28 Move
24 to the other side, becoming 24

2x = **28 24** Subtract
24 from 28

2x **/ 2** = 4 **/ 2** Divide both sides by 2 to get
the x value

x = 2

**x = 2**

**y = 4**

** **

** **

** **

** **

**Example III:** These equations have fractions in them, so it will be
a little harder due to the complications coming from combining them with
substitution, making things a bit confusing:

1. 6x + 2y = 2 4/5

2. 4x + 3y = 2 8/15

We must first, of course, centralize one variable. We will use y in the first equation:

**6x** + 2y = 2 4/5 Move
6x to the other side, becoming 6x

2y **/2** = (2 4/5
6x) **/ 2** Divide both sides by 2 to get the y value

y = 1 2/5 3x

Now substitute the new y value in the second equation and solve for x:

4x + 3**y** = 2
8/15 Substitute
(1 2/5 3x) for y

4x **+ 3(1 2/5 3x)**
= 2 8/15 Multiply out
the brackets by 3

**4x** + 4 1/5 ** 9x** = 2 8/15 Combine the y variables

-5x **+ 4 1/5** = 2 8/15 Move 4 1/5 to the other side, becoming 4 1/5

-5x = **2 8/15 4 1/5** Find a common denominator for 2 8/15 and 4 1/5

-5x = **2 8/15 4 3/15** Subtract the two terms

-5x **/ -5** = -1 2/3 **/ -5** Divide both
sides by -5 to get the value of x

x = 1/3

Place that value into one of the first two equations and solve for y. We will use the first:

6**x** + 2y = 2 4/5 Substitute 1/3 for x

**6(1/3)** + 2y = 2
4/5 Multiply
6 by 1/3

**2** + 2y = 2 4/5 Move 2 to the other side,
becoming 2

2y = **2 4/5 2 ** Combine
the two terms

2y **/ 2** = 4/5 **/ 2** Divide both sides by 2

y = 2/5

**x = 1/3**

**y = 2/5**

** **

**Example IV:** A regular set of equations, with a twist in the
answer. Can you guess what it is? Heres a hint - look at the third term of
the second equation:

1. 5x + 5y = 7

2. 10x 6y = 4.4

First get one variable alone. We will use x in the first equation:

5x **+ 5y** = 7 Move 5y to the other side, becoming 5y

5x **/ 5** = (7
5y) **/ 5** Divide both sides by 5 to get the x value

x = 1.4 y

Now substitute the new x value into the second equation:

10**x** 6y = 4.4 Substitute
(1.4 y) for x

**10(1.4 y)**
6y = 4.4 Multiply out
the brackets by 10

14 ** 10y 6y** =
4.4 Combine the y
variables

14 16y = 4.4 Move 14 to the other side, becoming 14

-16y = **4.4 14 ** Combine
the two terms

-16y **/ -16** =
-9.6 **/ -16** Divide both sides by 16 to get the y value

y = 0.6

Now substitute your new value into one of the first two equations. We will use the first:

5x + 5**y** = 7 Substitute 0.6 for y

5x + **5(0.6)** = 7 Multiply
5 by 0.6

5x **+ 3** = 7 Move
3 to the other side, becoming 3

5x = **7 3** Combine
the two terms

5x **/ 5** = 4 **/ 5** Divide both sides by 5 to get
the x value

x = 0.8

**x = 0.8**

**y = 0.6**

**Practice Questions**

** **

Here are seven questions to try on your own. The answers are in the appendix. The last is a word question that requires you to pick out the information and make your own equation.

1. x + y = 12

x y = 4

2. 2x + 4y = 26

4x + 2y = 34

3. 2x + 3y = 23

3x 4y = -8

4. 1/3x + 2y = 1 2/3

4x + 2/3y = 16 1/9

5. 1.4x 3y = -5.6

3x + 2.8y = 6.8

6. 7x + 3y = 3.425

6x y = - 0.1

7. John went to the store and bought 6 jars of pickles and 3 bottles of ketchup for $12.00. The next week he bought 4 jars of pickles and 5 bottles of ketchup for $12.50. How much does each product cost?

**Elimination**

Elimination is also used to solve for variables in equations, but involves (obviously) eliminating instead of substituting. The general idea is to combine the equations so that they eliminate one variable, allowing you to solve for the other. Our example equations will be:

1. 2x + 2y = 14

2. 4x 2y = 4

The first step is to combine the equations. This is done by combining the like terms into a new super-equation if you will. Ideally, one of the variables will have coefficients that will cancel each other out when added (ie. 5 and 5 will equal 0). If not, you need to play around with it a bit, but that will be explained later. Luckily, this equation is set up quite lovely and the y variables will cancel. So we combine:

2x + 2y = 14

__+____ 4x
2y = 4 __

6x + 0 = 18 or 6x = 18

The y variables leave an equation that we can easily solve:

6x **/ 6 **= 18 **/ 6** Divide both sides by 6 to get
the x value

x = 3

I must say this is a heck of a lot easier than substitution. Of course, only for certain equations. Complex equations will require a lot of work to combine as will come up later. Anyway, place your x value in one of your first two equations and solve for y. We will use the first:

2**x** + 2y = 14 Substitute
x for 3

**2(3)** + 2y = 14 Multiply
2 by 3

**6** + 2y = 14 Move 6 to the other side, becoming 6

2y = **14 6** Combine the terms

2y **/ 2** = 8 **/ 2** Divide both sides by 2 to get
the y value

y = 4

Remarkable eh? We should, to make sure we are absolutely correct, check our values by putting them into the original equations:

1. 2x + 2y = 14

2(3) + 2(4) = 14

6 + 8 = 14

14 = 14

2. 4x 2y = 4

4(3) 2(4) = 4

12 8 = 4

4 = 4

**Examples**

Now are some examples of equations we will solve with elimination, each increasing in difficulty and adding new twists to the process.

** **

**Example I:** These are fairly simple equations that require a bit of
tinkering before we can eliminate:

1. 3x + 4y = 27

2. 5x + 4y = 37

We cant eliminate yet because our variable coefficients will not cancel each other out. But luckily the y coefficients are the same value, so we can multiply one equation by 1, allowing us to successfully eliminate the y variable. We will use the first equation, as the values are smaller in the other terms and when added will leave us nice positive numbers.

(3x + 4y = 27)**(-1) ** Multiply the equation by 1

- 3x 4y = - 27

We can combine the equations and eliminate the y variables

-3x 4y = - 27

__+____ 5x
+ 4y = 37 __

2x + 0 = 10 or 2x = 10

Now we can solve for x with the super-equation:

2x **/ 2** = 10 **/ 2** Divide both sides by 2 to get
the x value

x = 5

Our x value can be placed into one of the first two equations and we can solve for y. We will use the first:

3**x** + 4y = 27 Substitute
5 for x

**3(5)** + 4y = 27 Multiply
3 by 5

**15** + 4y = 27 Move
15 to the other side, becoming 15

4y = **27 15** Combine
the terms

4y **/ 4** = 12 **/ 4 ** Divide both sides by 4 to get
the y value

y = 3

**x = 5**

**y = 3**

** **

**Example II: **Also
simple equations requiring a different kind of manipulation:

1. 2x + 2y = - 2

2. 4x + y = 34

In these equations, again our initial variable coefficients wont cancel each other out. But there is a light at the end of the tunnel. We have a negative, the 2x, which we can multiply by 2 so we can eliminate it with the 4x of our second equation. This will require the multiplication of the entire first equation by 2.

(- 2x + 2y = - 2)**(2)** Multiply the equation by 2

-4x + 4y = - 4

Now we can combine:

-4x + 4y = -4

__+____ 4x
+ y + 34__

0 + 5y = 30 or 5y = 30

The super-equation! The next logical course of action is to solve for y:

5y **/ 5** = 30 **/ 5** Divide both sides by 5 to get
the value of y

y = 6

The y value can be placed into one of the first two equations and we can find the value of x. We will use the second equation:

4x + **y** = 34 Substitute 6 for y

4x **+ 6** = 34 Move 6 to the other side, becoming 6

4x = **34 6** Combine the terms

4x **/ 4** = 28 **/ 4** Divide both sides by 7 to get
the value of x

x = 7

**x=7**

**y=6**

** **

**Example III:** We now progress into the question that requires our
astute mental prowess. These equations
will use both of the twists and manipulation of the first two examples:

1. 3x + 2y = 20

2. 6x + 3y = 33

Now what can we eliminate? Nothing right off hand. Can we multiply it by 1 to get something to eliminate? No, that wont do it. Can we multiply one equation by some term that will allow us to eliminate? No, that too is an unsatisfactory course of action. But can we combine those two suggestions, and multiply by a negative term so to eliminate? Why yes, we can. If we multiply 3x of the first equation by 2, we can eliminate it with 6z of the second equation, like so:

(3x + 2y = 20)**(-2)** Multiply the equation by
-2

-6x 2y = - 20

And then we combine the equations:

-6x 2y - = - 20

__+____ 6x
+ 3y = 33__

0 + 1y = 7 or y = 7

To our great luck, there is no need to solve for y as our combining does so for us, so we skip that bit and place our y value into one of the first two equations. We shall use the first:

3x + 2**y** = 20 Substitute
7 for y

3x + **2(7)** = 20 Multiply
2 by 7

3x **+ 14** = 20 Move
14 to the other side, becoming 14

3x = **20 14** Combine
the terms

3x **/ 3** = 6 **/ 3** Divide both sides by 3 to get
the x value

x = 2

**x= 2**

**y = 7**

** **

** **

** **

** **

** **

**Example IV: **These
equations have absolutely nothing in common.
Should be fun:

1. 7x + 2y = 25

2. 3x + 9y = 27

To solve these equations we must multiply both equations by something that will allow us to eliminate, as the possibility of an easy solution seems to have eluded us this time. So what will we use? Well, weve been eliminating x the last few times first, so lets eliminate the y. The simple route is to multiply the equation with 2y by 9 and the equation with 9y by 2. Also, one should be negative to allow us to eliminate and we will go with the 2 so as to keep our positive numbers when we combine, which is just easier. You may think this is a rather random way to solve this. Yeah. Thats what you gotta do with no clear options:

(7x + 2y = 25)**(9)** Multiply
the equation by 9

63x + 18y = 225

(3x + 9y = 27)**(-2)** Multiply the equation by -2

-6x 18y = - 54

combine the two new equations:

63x + 18y = 225

__+____ -
6x 18y = - 54__

57x + 0 = 171 or 57x = 171

Now we solve for x:

57x **/ 57** = 171 **/ 57** Divide
both sides by 57 to get the x value

x = 3

Place the new x value into one of the first two equations and solve for x. We will use the first:

7**x** + 2y = 25 Substitute
x with 3

**7(3)** + 2y = 25 Multiply
7 by 3

**21** + 2y = 25 Move
21 to the other side, becoming 21

2y = **25 21** Combine
the terms

2y **/ 2** = 4 **/ 2** Divide both sides by 2 to get
the y value

y = 2

**x = 3**

**y = 2**

** **

**Practice Questions**

** **

Here are seven questions to try using elimination. The answers will be in the appendix:

1. x + y = 12

x y = - 4

2. .4x 2y = 6

3x + 2y = 22

3. 2x + 3y = 19

2x + 2y = 14

4. 4x 7y = 10

5x + 14y = 58

5. -3x 2y = - 18

-4x 7y = - 37

6. 3x y = 5n

4x + y = 9n

7. Paul went to the store and bought two records and a poster for $23.99. The next month he bought one record and 4 posters for $43.46. What was the cost of each product?

** **

** **

** **

** **

** **

** **

** **

**Elimination With Three Variables**

The process for eliminating with three variables is similar to that of two, but with some additional steps. Our explanatory equations will be:

1. 2x + 3y + 2z = 20

2. 3x 3y + 4z = 19

3. 4x + 3y 2z = 10

The first step is to move from three equations to two. How? you may ask. Well, we must take two equations and eliminate one variable, then take two other equations (obviously one will be an equation used in the first elimination) and eliminate the same variable. From there we pick up our old eliminating techniques we just learned. So what will we eliminate? Well, we have a -3y and two 3y in various equations, so we will eliminate the 3y equations with the -3y and go from there. Funny, the luck we had with that like it was all arranged to work out. This sounds confusing but should make sense momentarily:

2x + 3y + 2z = 20

__+____ 3x
3y + 4z = 19__

5x + 0 + 6z = 39 or 5x + 6z = 39

4x + 3y 2z = 10

__+____ 3x
3y + 4z = 19__

7x + 0 + 2z = 29 or 7x + 2z = 29

These become our new equations:

4. 5x + 6z = 39

5. 7x + 2z = 29

Now we eliminate with these to get our x and z values. Unfortunately, there is no clear elimination, so we must play with it a bit. It looks like if we multiply the fifth equation by 3 we can cancel the z variables. In three variable elimination, rarely is the two variable part spelled out for you. Anyway:

(7x + 2z = 29)**(-3) ** Multiply the equation by 3

-21x 6z = - 87

Now we combine with equation four:

5x + 6z = 39

__+____
-21x 6z = - 87__

-16x + 0 = - 48 or 16x = - 48

And solve for x:

- 16x **/ -16** = -
48 **/ -16** Divide
both sides by 16 to get the x value

x = 3

With our new x value, we can solve for z by using equation four or five. Well use four:

5**x** + 6z = 39 Substitute
3 for x

**5(3)** + 6z = 39 Multiply
5 by 3

**15** + 6z = 39 Move
15 to the other side, becoming 15

6z =** 39 15** Combine
the terms

6z **/ 6** = 24 **/ 6** Divide both sides by 6 to get
the z value

z = 4

Are we done? No, we still have to get our y value by placing the two values we found in one of the first three equations and solving for y. We will use the first equation:

2**z** + 3y + 2z =
20 Substitute
3 for x

2(3) + 3y + 2**z**
= 20 Substitute 4 for z

**2(3)** + 3y + **2(4)** = 20 Multiply out both brackets

**6** + 3y **+ 8** = 20 Combine
like terms

3y **+ 14** = 20 Move
14 to the other side, becoming 14

3y = **20 14** Combine
the terms

3y **/ 3** = 6 **/ 3** Divide both sides by 3 to get
the y value

y = 2

Our answer will be:

**x = 3**

**y = 2**

**z = 4**

Of course, we should check our answers. We will put our values into the second and third equations to make sure it works out. a good habit to form:

2. 3x 3y + 4z = 19

3(3) 3(2) + 4(4) = 19

9 6 + 16 = 19

19 = 19

3. 4x +3y 2z = 10

4(3) + 3(2) 2(4) = 10

12 + 6 8 = 10

10 = 10

**Examples**

**Example 1:** Usually we start with the super easy question here, but
instead it will be a regular question.
It will need some work before we can eliminate:

1. x + 2y + 3z = 29

2. 2x 3y + 4z = 25

3. 4x + 5y + 5z = 25

Now what variable should we eliminate? It must be the same for both new equations. X seems to be the best option. Equation one can be multiplied by 2 to eliminate with equation two, and equation two can be doubled to eliminate the x with equation three. We will fix equation one:

(x + 2y + 3z = 29)**(-2)** Multiply
the equation by 2

-2x 4y 6z = - 58

Combine with the second equation:

-2x 4y 6z = - 58

__+____ 2x
3y + 4z = 25__

0 7y 2z = - 33 or 7y + 2z = 33

Now we doctor equation two:

(2x 3y + 4z = 25)**(2)** Multiply
the equation by 2

4x 6y + 8z = 50

And eliminate with equation three:

- 4x + 5y + 5z = 25

__+____ 4x
6y + 8z = 50__

0 y + 13z = 75 or y + 13z = 75

Our two new equations are:

4. 7y + 2z + 33

5. - y + 13z = 75

Now this is a fairly good spot to be in. We can eliminate by multiplying the fifth equation by 7, allowing for the cancellation of the y variable:

(- y + 13z = 75)**(7)** Multiply
the equation by 7

- 7y + 91z = 525

And combine with equation four:

- 7y + 91z = 525

__+____ 7y
+ 2z = 33__

0 + 93z = 558 or 93z = 558

Solve for z:

93z **/ 93** = 558 **/ 93** Divide
both sides by 92 to get the z value

z = 6

Now we take our z value and place it into either our fourth or fifth equation to solve for y. We will use the fourth:

7y + 2**z** = 33 Substitute
6 for z

7y **+ 2(6)** = 33 Multiply
2 by 6

7y **+ 12** = 33 Move
12 to the other side, becoming 12

7y = **33 12** Combine
the terms

7y **/ 7** = 21 **/ 7** Divide both sides by 7 to get
the v value

y = 3

Of course, now we take our z and y values and place them into one of the first three equations and solve for x. We will use the first:

x + 2**y** + 3z =
29 Substitute 3 for y

x + 2(3) + 3**z** =
29 Substitute 6 for z

x **+ 2(3) + 3(6)**
= 29 Multiply out the
brackets

x **+ 6 + 18** = 29 Combine
like terms

x **+ 24** = 29 Move 24 to the other side, becoming 24

x = **29 24** Combine the terms

x = 5

And the answer is:

**x = 5 y=3
z=6**

**Example 2:**. These equations
will have nothing in common and more than likely result in large numbers as we
try to solve:

1. 12x 5y + 6z = 13

2. 14x + 8y 5z = 5

3. 18x 3y 4z = - 2

What do we have to eliminate? Nothing. Anything we can double or turn into a negative? Nope. So what can we do? The tried and true method of trial and error. What variable will we eliminate? Lets say z as it has yet to be eliminated at this stage in our previous questions. Z actually has a half-decent set up. We can multiply the 6z by 5 and the 5z by 6 and eliminate there. Also, we can multiply the 6z by 4 and the 4z by 6 and eliminate there:

(12x 5y + 6z = 13)**(5) ** Multiply
the equation by 5

60x 25y + 30z = 65

(14x + 8y 5z = 5)**(6)** Multiply
the equation by 6

84x + 48y 30z = 30

60x 25y + 30z = 65

__+ ____84x
+ 48y 30x = 30__

144x + 23y + 0 = 95 or 144x + 23y = 95

And get the other equation:

(12x 5y + 6z = 13)**(4)** Multiply
the equation by 4

48x 20y + 24z = 52

(18x 3y 4z = - 2)**(6) **Multiply
the equation by 6

108x 18y 24z = - 12

108x 18y 24z = - 12

__+____ 48x
20y + 24z = 52__

156x 38y + 0 = 40 or 156z 38y = 40

Our two new equations are:

4. 144x + 23y = 95

5. 156x 38y = 40

Bad setup. We will eliminate the y variables now, as they are lower and will keep our numbers smaller and easier to work with. Well multiply equation four by 38 and equation five by 23, so as to be able to cancel the variables:

(144x + 23y = 95)**(38)** Multiply
the equation by 38

5472x + 874y = 3610

(156x 38y = 40)**(23)** Multiply
the equation by 23

3588x - 874y = 920

And we eliminate:

5472x + 874y = 3610

__+____
3588x - 874y = 920__

9060x 0 = 4530 or 9060x = 4530

Now solve for x:

9060x **/ 9060** = 4530 **/ 9060** Divide both
sides by 9060 to get the x value

x = 0.5

And now we place that into equation four and five and solve for y. Well use number four:

144**x** + 23y = 95 Substitute
0.5 for x

**144(0.5)** + 23y
= 95 Multiply 144 by 0.5

**72** + 23y = 95 Move
72 to the other side, becoming 72

23y = **95 72** Combine
the terms

23y **/ 23** = 23 **/ 23** Divide
both sides by 23 to get the y value

y = 1

This stuff takes forever. Be very careful when doing your math one sign mixed up and the whole thing is shot. Concentration is the word of the day. Now to solve for z with one of the first three equations. Well use the first:

12**x** 5y + 6z =
13 Substitute
0.5 for x

12(0.5) 5**y** +
6z = 13 Substitute 1 for y

**12(0.5) 5(1)**
+ 6z = 13 Multiply out
the brackets

**6 5** + 6z = 13 Combine like terms

6z **+ 1** = 13 Move
1 to the other side, becoming 1

6z = **13 1** Combine
the terms

6z **/ 6** = 12 **/ 6** Divide both sides by 6 to get the z value

z = 2

Our answer is:

**x = 0.5**

**y = 1**

**z = 2**

** **

**Practice Questions**

** **

Here are three questions for you to try on elimination with three variables. Enjoy!

1. 2x + 3y + 4z = 29

3x + 2y 4z = - 4

5x + 4y 4z = 6

2. x + y + 2z = 15

4x 2y + z = 22

3x 4y + 5z = 15

3. 4x + 5y + 8z = 301

3x + 2y z = 109

5x 9y + 5z = 12

**Closing Notes**

When dealing with substitution and elimination, there are a few key points to remember for both. First, questions can be hidden in many forms. They can appear like our word questions, but are also sneaked into questions involving phone plans or coin amounts. The key when reading a question is to see if you can get similar variables and a value for it to equal. Then youre on your way. Another thing is that when looking at your equations, for substitution, on that first step try to use the equation that divides easiest. It makes the rest so much easier. For elimination, when multiplying try to keep the numbers positive and as small as possible, also for ease of use. And avoid fractions or decimals they were created solely to confuse you. But the key thing is to pay attention when you do these questions. One little mistake will mess up everything else (and I speak from experience making this kit). And always check your values at the end, even if just in your head. Always.

**Quiz**

** **

Youve made it through the tutorial part, now its time to test your newfound skills in the quiz. Good luck!

**1.
****Solve the following
with substitution:**

a. 2x + 3y = 17

x + 2y = 10

b. 3x + 6y = 36

4x + 2y = 30

c. 2x + 4y = - 6

3x 2y = 23

d. 1/3x + 3y = 28

2x + 2/3y = 12

**2.
****Solve the following
using elimination:**

a. 5x + 3y = 25

4x 3y = -7

b. 2x + 5y = 46

2x 4y = -26

c. 5x 3y = 26

3x + 6y = 0

d. 9x + 8y = 9.5

5x + 6y = 5.9

e. 4x + 2y z = 17

3x 2y + 4z = 6

x 2y + 9z = 17

f. 2x 4y + 3z = - 29

4x 6y + 5z = - 41

5x 8y + 7z = - 55

3. George goes to the store one day and buys a box of Cheerios and two cartons of milk for $6.50. The next week he buys two boxes of Cheerios and one carton of milk for $7.00. How much does each item cost?

4. Herman, desiring a hipper look, goes to trendy clothing store and buys three pairs of bellbottoms and a Nehru jacket for $65. His friend Gerry, seeing how cool Herman now looks, goes to the same store and buys two pairs of bellbottoms and two Nehru jackets for $70. How much does each item cost?

5. Johnny, having little luck with the ladies, decides to change his hairstyle and odor so as to impress them. He buys two bottles of shampoo, three bottles of hair gel and one bottle of cologne for $24. The next month, having success with his new endeavor, he buys one bottle of shampoo, two bottles of hair gel, and two bottles of cologne for $24. The next month his results are so good that he buys two more bottles of shampoo, three of hair gel, but no cologne as he had some left over, for $17. How much does each product cost?

**Appendix**

** **

**Substitution:**

1. x + y = 12

x y = 4 = x = y + 4

(y + 4) + y = 12

y + 4 + y = 12

2y = 8

**y = 4**

x 4 = 4

**x = 8**

2. 2x + 4y = 26 = x = - 2y + 13

4x + 2y = 34

4(-2y + 13) + 2y = 34

-8y + 52 + 2y = 34

-6y = -18

**y = 3**

2x + 4(3) = 26

2x + 12 = 26

2x = 14

**x = 7**

3. 2x + 3y = 23

3x 4y = -8 = y = 0.75x + 2

2x + 3(0.75x +2) = 23

2x + 2.25x + 6 = 23

4.25x = 17

**x = 4**

** **

3(4) 4y = -8

12 4y = -8

-4y = -20

**y = 5**

4. 1/3x + 2y = 1 2/3 = x = - 6y + 5

4x + 2/3y = 16 1/9

4(-6y +5) + 2/3y = 16 1/9

-24y + 20 + 2/3y = 16 1/9

-23 1/3y = - 3 8/9

**y = 1/6**

1/3x + 2(1/6) = 1 2/3

1/3x + 1/3 = 1 2/3

1/3x = 1 1/3

**x = 4**

5. x 3y = -5.6 = x = 3y 5.6

3x + 2.8y = 6.8

3(3y 5.6) + 2.8y = 6.8

9y 16.8 + 2.8y = 6.8

11.8y = 23.6

**y = 2**

x 3(2) = -5.6

x 6 = -5.6

**x = 0.4**

6. 7x + 3y = 3.425

6x y = - 0.1 = y = 6x + 0.1

7x + 3(6x + 0.1) = 3.425

7x + 18x + 0.3 = 3.425

25x = 3.125

**x = 0.125**

** **

6(0.125) y = - 0.1

0.75 y = - 0.1

**y = 0.85**

7. 6p + 3k = 12 = k = - 2p + 4

4p + 5k = 12.50

4p + 5(-2p + 4) = 12.5

4p 10p + 20 = 12.5

-6p = - 7.5

**p = 1.25**

** **

6(1.25) + 3k = 12

7.5 + 3k = 12

3k = 4.5

** k = 1.5**

** **

**Elimination:**

1. x + y = 12

x y = - 4

x + y = 12

__+____ x y = -4__

2x 0 = 8

2x = 8

**x = 4**

4 + y = 12

**y = 8**

2. 4x 2y = 6

3x + 2y = 22

4x 2y = 6

__+ ____3x + 2y = 22__

7x + 0 = 28

7x = 28

**x = 4**

** **

3(4) + 2y = 22

12 + 2y = 22

2y = 10

**y = 5**

3. 2x + 3y = 19

2x + 2y = 14(-1) = - 2x 2y = - 14

- 2x 2y = - 14

__+____ 2x + 3y = 19__

0 + y = 5

**y = 5**

** **

2x + 3(5) = 19

2x + 15 = 19

2x = 4

**x = 2**

4. 4x 7y = 10(2) = 8x 14y = 20

5x + 14y = 58

8x 14y = 20

__+ ____ 5x + 14y = 58__

13x + 0 = 78

13x = 78

**x = 6**

** **

5(6) + 14y = 58

30 + 14y = 58

14y = 28

**y = 2 **

5. -3x 2y = - 18(7) = - 21x 14y = - 126

-4x 7y = - 37(-2) = 8x + 14y = 74

-21x 14y = -126

__+ ____8x + 14y = 74__

-13x + 0 = -52

-13x = -52

**x = 4**

-3(4) 2y = -18

-12 2y = -18

-2y = =6

**y = 3**

6. 3x y = 5n

4x + y = 9n

3x y = 5n

__+____ 4x + y = 9n__

7x + 0 = 14n

7x = 14n

**x = 2n**

4(2n) + y = 9n

8n + y = 9n

**y = n**

7. 2r + p = 23.99

r + 4p = 43.46(-2) = -2r 8p = - 86.92

- 2r 8p = - 86.92

__+____ 2r + p = 23.99__

0 7p = - 62.93

-7p = -62.93

**p = 8.99**

r + 4(8.99) = 43.46

r + 35.96 = 43.46

**r = 7.50**

** **

**3x3 Elimination:**

1. 2x + 3y + 4z = 29

4x + 2y 4z = - 2

5x + 4y 4z = 6

2x + 3y + 4z = 29

__+____ 4x + 2y 4z = - 2__

6x + 5y + 0 = 27

2x + 3y + 4z = 29

__+ ____5x + 4y 4z = 6__

7x + 7y + 0 = 35

6x + 5y = 27(7) = 42x + 35y = 189

7x + 7y = 35(-5) = - 35x 35y = - 175

- 35x 35y = - 175

__+ ____42x + 35y = 189__

7x + 0 = 14

7x = 14

**x = 2**

** **

6(2) + 5y = 27

12 + 5y = 27

5y = 15

**y = 3**

2(2) + 3(3) + 4z = 29

4 + 9 + 4z = 29

4z = 16

**z = 4**

2. x + y + 2z = 15(2) = 2x + 2y + 4z = 30

4x 2y + z = 22(-2) = - 8y + 4y 2z = - 44

3x 4y + 5z = 15

2x + 2y + 4z = 30

__+____ 4x 2y + z = 22__

6x + 0 + 5z = 52

- 8y + 4y 2z = - 44

__+____ 3x 4y + 5z = 15__

-5x + 0 + 3z = -29

-5x + 3z = -29(6) = - 30x + 18z = - 174

6x + 5z = 52(5) = 30x + 25z = 260

- 30x + 18z = - 174

__+____ 30x + 25z = 260__

0 + 43z = 86

43z = 86

**z = 2**

** **

6x + 5(2) = 52

6x + 10 = 52

6x = 42

**x = 7**

** **

7 + y + 2(2) = 15

7 + y + 4 = 15

y + 11 = 15

**y = 4**

3. 4x + 5y + 8z = 301

3x + 2y z = 49(8) = 24x + 16y 8z = 392

5x 9y + 5z = 12

24x + 16y 8z = 392

__+____ 4x + 5y + 8z = 301__

28x + 21y + 0 = 693

3x + 2y z = 49(5) = 15x + 10y 5z = 245

15x + 10y 5z = 245

__+____ 5x 9y + 5z = 12__

20x + y + 0 = 257

28x + 21y = 693

20x + y = 257(-21) = - 420x 21y = -5397

- 420x 21y = -5397

__+____ 28x + 21y = 693__

-392x + 0 = - 4704

-392x = - 4704

**x = 12**

** **

28(12) + 21y = 693

336 + 21y = 693

21y = 357

**y = 17**

4(12) + 5(17) + 8z = 301

48 + 85 + 8z = 301

8z = 168

**z = 21**

** **

**Quiz:**

** **

1.
** Solve the following with substitution:**

a. 2x + 3y = 17

x + 2y = 10 = x = - 2y + 10

2(-2y + 10) + 3y = 17

-4y + 20 + 3y = 17

-y = -3

**y = 3**

2x + 3(3) = 17

2x + 9 = 17

2x = 8

**x = 4**

b. 3x + 6y = 36 = x = - 2y + 12

4x + 2y = 30

4(-2y + 12) + 2y = 30

-8y + 48 + 2y = 30

-6y = -18

**y = 3**

** **

3x + 6(3) = 36

3x + 18 = 36

3x = 18

**x = 6**

c. 2x + 4y = - 6 = x = - 2y - 3

3x 2y = 23

3(-2y 3) 2y = 23

-6y 9 2y = 23

-8y = 32

**y = - 4**

** **

2x + 4(-4) = - 6

2x 16 = -6

2x = 10

**x = 5**

d. 1/3x + 3y = 28

2x + 2/3y = 12 = x = - 1/3y + 6

1/3(-1/3y + 6) + 3y = 28

- 1/9y + 2 + 3y = 28

2 8/9y = 26

**y = 9**

** **

2x + 2/3(9) = 12

2x + 6 = 12

2x = 6

**x = 3**

**2. ****Solve the following using elimination:**

a. 5x + 3y = 25

4x 3y = -7

5x + 3y = 25

__+____ 4x 3y = -7__

9x + 0 = 18

9x = 18

**x = 2**

** **

5(2) + 3y = 25

10 + 3y = 25

3y = 15

**y = 5**

b. 2x + 5y = 46

2x 4y = -26(-1) = - 2x + 4y = 26

- 2x + 4y = 26

__+ ____2x + 5y = 46__

0 + 9y = 72

9y = 72

**y = 8**

2x + 5(8) = 46

2x + 40 = 46

2x = 6

**x = 3**

c. 5x 3y = 26(2) = 10x 6y = 52

3x + 6y = 0

10x 6y = 52

__+ ____3x + 6y = 0__

13x + 0 = 52

13x = 52

**x = 4**

** **

3(4) + 6y = 0

12 + 6y = 0

6y = - 12

**y = - 2**

d. 9x + 8y = 9.5(-5) = - 45 40y = - 47.5

5x + 6y = 5.9(9) = 45x + 54y = 53.1

- 45 40y = - 47.5

__+ ____45x + 54y = 53.1__

0 + 14y = 5.6

14y = 5.6

**y = 0.4**

** **

9x + 8(0.4) = 9.5

9x + 3.2 = 9.5

9x = 6.3

**x = 0.7**

e. 4x + 2y z = 17

3x 2y + 4z = 6

x 2y + 9z = 17

4x + 2y z = 17

__+____ 3x 2y + 4z = 6__

7x + 0 + 3z = 23

4x + 2y z = 17

__+____ x 2y + 9z = 17__

5x + 0 + 8z = 34

7x + 3z = 23(-5) = -35x 15z = - 115

5x + 8z = 34(7) = 35x + 56z = 238

-35x 15z = - 115

__+____ 35x + 56z = 238__

0 + 41z = 123

41z = 123

**z = 3**

7x + 3(3) = 23

7x + 9 = 23

7x = 14

**x = 2**

** **

4(2) + 2y 3 = 17

8 + 2y 3 = 17

2y = 12

**y = 6**

** **

f. 2x 4y + 3z = - 29(5) = 10x 20y + 15z = - 145

4x 6y + 5z = - 41(-3) = - 12x + 18y 15z = 123

5x 8y + 7z = - 55

- 12x + 18y 15z = 123

__+ ____10x 20y + 15z = - 145__

-2x 2y + 0 = - 22

4x 6y + 5z = - 41(7) = 28x 42y + 35z = - 287

5x 8y + 7z = - 55(-5) = - 25x + 40y 35z = 275

- 25x + 40y 35z = 275

__+ ____28x 42y + 35z = - 287__

3x 2y + 0 = -12

-2x 2y = - 22

3x 2y = -12(-1) = - 3x + 2y = + 12

-2x 2y = - 22

__+ ____- 3x + 2y = +12__

-5x + 0 = - 10

-5x = - 10

**x = 2**

** **

3(2) 2y = - 12

6 2y = -12

-2y = - 18

**y = 9 **

** **

5(2) 8(9) + 7z = - 55

10 72 + 7z = - 55

7z = 7

**z = 1**

3. c + 2m = 6.50 = c = -2m + 6.50

2c + m = 7.00

2(-2m + 6.50) + m = 7

- 4m + 13 + m = 7

- 3m = -6

**m = 2**

2c + 2 = 7

2c = 5

**c = 2.50**

4. 3b +n = 65(-2) = -6b 2n = - 130

2b + 2n = 70

-6b 2n = - 130

__+ ____2b + 2n = 70__

-4b + 0 = -60

-4b = -60

**b = 15**

** **

2(15) + 2n = 70

30 + 2n = 70

2n = 40

**n = 20**

5. 2s + 3h + c = 24(-2) = -4s 6h 2c = - 48

s + 2h + 2c = 24

2s + 3h = 24

-4s 6h 2c = - 48

__+ ____s + 2h + 2c = 24__

- 3s 4h + 0 = - 24

- 3s 4h = - 24(2) = - 6s 8h = - 48

2s + 3h = 17(3) = 6s + 9h = 51

- 6s 8h = - 48

__+____ 6s + 9h = 51__

0 + h = 3

**h = 3**

** **

2s + 3(3) = 17

2s + 9 = 17

2s = 8

**s = 4**

** **

4 + 2(3) + 2c = 24

4 + 6 + 2c = 24

2c = 14

**c = 7**

**Web Sites For Additional Information**

** **

Assuming you need additional info after this amazing tutorial, here are a few pages to look up:

www.askdrmath.com A good page for any math question. The archives contain tons of info on pretty much any math related question.

www.sosmath.com Another good math page. With very detailed information. I highly recommend it.

www.coolmath.com A site designed for easier math things, but a bit of a help. Recommended for the games mostly, especially Lunar Lander.

**Bibliography**