The green rectangles will
represent positive x. The
red rectangles will equal
negative x. One of each
will equal a neutral pair
and cancel out to zero.
The purple rectangles will
represent positive y. The
black rectangles will equal
negative y. One of each
will equal a neutral pair
and cancel out to zero.
The yellow squares will
represent positive 1. The
red squares will equal
negative 1. One of each
will equal a neutral pair
and cancel out to zero.
Substitution method for more complicated
problems may require you to use the distributive
property as you solve for a second variable.
You do not need to always solve for y. You can
choose to solve for x instead of putting in into
y = mx+b form.
Ex. 1
Solve the system of equations:
2x + 4y = - 4
4x - 2y = 12
=
=
You can pick either equation and use it to isolate for either variable. Try to choose the
one that will be the least amount of work for yourself.
I choose to solve for x in the first equation.
Ex. 1
Solve the system of equations:
2x + 4y = - 4
4x - 2y = 12
=
=
=
After you isolate for x, you
need to plug that value into
the other equation.
Ex. 1
Solve the system of equations:
2x + 4y = - 4
4x - 2y = 12
After you isolate for x, you need to plug
that value into the other equation.
=
=
=
-8 -10y = 12
Ex. 1
Solve the system of equations:
2x + 4y = - 4
4x - 2y = 12
=
-10y = 20
=
=
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
=
=
You can pick either equation and use it to isolate for either variable. Try to choose the
one that will be the least amount of work for yourself.
I choose to solve for y in the second equation.
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
=
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
=
2 purple ones are equal to 8 yellow ones and 4 greens ones.
The means that 1 purple one is equal to 4 yellows and 2 greens ones.
=
You will need to take the value you just got for y, and replace it for the y’s that were in the other
equation.
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
=
=
Take what you just got for y, and replace
every y in the other equation with the
equivalent expression.
Then find out how much a green tile is
worth.
=
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
You need to isolate the green x terms onto one side and the constant
yellow unit terms onto the other side. I add negative unit squares to the
left side to make neutral pairs. Then I must add the same thing to the right
side
=
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
You now have 12x = 4.
What does x need to be in order for 12 of them
to be equal to 4?
Each green tile will be equal to 1/3 of a red tile.
=
Ex. 2
Solve the system of equations:
6x + 3y = 9
-4x + 2y = 8
You now have 12x = 4.
What does x need to be in order for 12 of them
to be equal to 4?
Each green tile will be equal to 1/3 of a red tile.
=

1
x
3
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
=
=
=
=
=
=
1
1
x   and y  3
3
3
 1 1 
 ,3 
 3 3 
Ex. 2
Solve the system of equations:
6x + 3y = 8
-4x + 2y = 8
Plug in both of the x an y values into each of the two
original equations to make sure your answers work.
6x  3y  8
 1   1 
Does 6  33  8?
 3   3 
Does 2  10  8?
8  8 True
4 x  2y  8
 1   1 
Does  4  23  8?
 3   3 
 1   2 
Does 1  6  8?
 3   3 
8  8 True