Solving Systems
of Equations by
Substitution
by
Tammy Wallace
Varina High School
Solving by Systems by Substitution
Another method for solving system of equations is the
substitution method.
This is done by replacing one variable (y) with an
equivalent expression that contains the other variable
(x).
From there, a one-variable equation is created, that
will be used to find the solution set.
𝑦 = 3𝑥 − 1
Find the solution set for 7𝑥 + 2𝑦 = 37
Procedures
Solve one equation for one
variable.
NOTE: Either equation can be chosen
Which equation did you
choose?
𝑦 = 3𝑥 − 1
7𝑥 + 2𝑦 = 37
Because this equation
is already solved for
y, it is easier to use
this one.
𝑦 = 3𝑥 − 1
______________________
While solved for a variable,
substitute the equation above
into the second equation and
solve for the remaining
variable.
What did that variable equal?
x=3
______________________
7x + 2y = 37
7x + 2(3x - 1) = 37
7x + 6x – 2 = 37
13x – 2 = 37
+2 +2
13x = 39
13
13
x=3
𝑦 = 3𝑥 − 1
Find the solution set for 7𝑥 + 2𝑦 = 37
Procedures
Substitute the value of
the variable above into
one of the original
equations to solve for
the remaining unknown
variable.
What did that variable
equal?
y=8
______________________
𝑦 = 3𝑥 − 1
7𝑥 + 2𝑦 = 37
Remember x = 3
y = 3x – 1
y = 3(3) – 1
y=9–1
y=8
𝑦 = 3𝑥 − 1
Find the solution set for 7𝑥 + 2𝑦 = 37
Procedures
𝑦 = 3𝑥 − 1
7𝑥 + 2𝑦 = 37
a) What is/are the solutions to the system? If x = 3 and y = 8, the
solution set is (3, 8)
b) If graphed, what type of lines would this system form and how can
you determine this WITHOUT graphing the system?
The graph would form intersecting lines because there is ONE
SOLUTION to the system.
c) Solve both equations for y and graphing with your calculator. Is the
solution set correct? YES!
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
Solve one equation for one
variable.
NOTE: Either equation can be chosen
Which equation did you
choose?
4𝑥 + 𝑦 = 3
−𝐱 − 𝟐𝐲 = 𝟖
4x + y = 3
-4x
-4x
y = -4x + 3
4𝑥 + 𝑦 = 3
______________________
While solved for a variable,
substitute the equation above
into the second equation and
solve for the remaining
variable.
What did that variable equal?
x=2
______________________
-x – 2y = 8
-x – 2(-4x + 3) = 8
-x + 8x – 6 = 8
7x – 6 = 8
+ 6 +6
7x = 14
7
7
x=2
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
Substitute the value of
the variable above into
one of the original
equations to solve for
the remaining unknown
variable.
What did that variable
equal?
y = −5
______________________
𝟒𝒙 + 𝒚 = 𝟑
−𝒙 − 𝟐𝒚 = 𝟖
Remember x = 2
-x – 2y = 8
-2 – 2y = 8
+2
+2
-2y = 10
-2
-2
y = -5
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
𝟒𝒙 + 𝒚 = 𝟑
−𝒙 − 𝟐𝒚 = 𝟖
a) What is/are the solutions to the system? If x = 2 and y = -5, the
solution set is (2, -5_)
b) If graphed, what type of lines would this system form and how can
you determine this WITHOUT graphing the system?
The graph would form intersecting lines because there is ONE
SOLUTION to the system.
c) Solve both equations for y and graphing with your calculator. Is the
solution set correct? YES!
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
Solve one equation for one
variable.
NOTE: Either equation can be chosen
Which equation did you
choose?
−𝑥 − 2𝑦 = 8
4𝑥 + 𝑦 = 3
−𝐱 − 𝟐𝐲 = 𝟖
-x – 2y = 8
+ 2y = +2y
-1(-x = 2y + 8 )
x = -2y - 8
______________________
While solved for a variable,
substitute the equation above
into the second equation and
solve for the remaining
variable.
What did that variable equal?
𝑦 = −5
______________________
4x + y = 3
4(-2y – 8) + y = 3
-8y – 32 + y = 3
-7y – 32 = 3
+32 +32
-7y = 35
-7
-7
y = -5
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
Substitute the value of
the variable above into
one of the original
equations to solve for
the remaining unknown
variable.
What did that variable
equal?
𝑥=2
______________________
𝟒𝒙 + 𝒚 = 𝟑
−𝒙 − 𝟐𝒚 = 𝟖
Remember y = -5
4x + y = 3
4x + (-5) = 3
4x – 5 = 3
+ 5 +5
4x = 8
4
4
x=2
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
𝟒𝒙 + 𝒚 = 𝟑
−𝒙 − 𝟐𝒚 = 𝟖
a) What is/are the solutions to the system? If x = 2 and y = -5, the
solution set is (2, -5_)
b) What do you notice about the solution to this system and solution to
problem # 2? What can you conclude about the process used their final
answers?
The solutions are the same. It doesn’t matter which equation
you solve for first and what variable you solve for first. As long
as the procedure is done correctly, the solution will be the same.
𝑥 =3−𝑦
Find the solution set for 𝑥 + 𝑦 = 7
Procedures
Solve one equation for one
variable.
NOTE: Either equation can be chosen
Which equation did you
choose?
𝑥 =3−𝑦
𝑥 =3−𝑦
𝒙+𝒚 =𝟕
Because this
equation is already
solved for y, it is
easier to use this
one.
______________________
While solved for a variable,
substitute the equation above
into the second equation and
solve for the remaining
variable.
x+y=7
3–y+y=7
3 =7
What happened when solving the equation for y? The variable cancelled leaving 3 = 7.
What can we conclude about the solution? Because 3 can NEVER equal 7, there
are NO SOLUTIONS.
What type of lines will the graph be? The lines are parallel.
2𝑥 + 𝑦 = 4
Find the solution set for 4𝑥 + 2𝑦 = 8
Procedures
5) Solve the system of equation
2𝑥 + 𝑦 = 4
4𝑥 + 2𝑦 = 8
While solved for a variable,
substitute the equation above into
the second equation and solve for
the remaining variable.
2𝑥 + 𝑦 = 4
2x + y = 4
-2x
-2x
y = -2x + 4
4𝑥 + 2𝑦 = 8
4x + 2y = 8
4x + 2(-2x + 4) = 8
4x – 4x + 8 = 8
8=8
What happened when solving the equation for y? Both sides of the equation are equal
What can we conclude about the solution? There are infinite many solutions.
What type of lines will the graph be? The lines are coinciding.
Solving Systems
of Equations by
Elimination
by
Tammy Wallace
Varina High School
Solving by Systems by Elimination
The Addition and Subtraction Properties of Equality
can be used solve a system of equation. Using this
method is called the ELIMINATION METHOD.
This is done by adding or subtracting the equation
together to eliminate one variable.
From there, the remaining variable is solved for a
specific value, which is then used to find the complete
solution to the system.
5𝑥 − 6𝑦 = −32
Find the solution set for 3𝑥 + 6𝑦 = 48
Procedures
Make sure both equation are in
Standard Form:
𝐀𝐱 + 𝐁𝐲 = 𝐂
(_____________)
While in standard form, which
terms has the same coefficient?
−𝟔𝐲 𝐚𝐧𝐝 𝟔𝐲
Including the operation in front
of each term, what operation
would cancel out those terms?
𝐀𝐝𝐝𝐢𝐭𝐢𝐨𝐧
Use this operation to eliminate
the terms . Solve for the
remaining variable.
The remaining variable equal?
𝐱=𝟐
𝟓𝒙 − 𝟔𝒚 = −𝟑𝟐
𝟑𝒙 + 𝟔𝒚 = 𝟒𝟖
Both equations are already in
Standard From
5x – 6y = -32
+ 3x + 6y = 48
8x
= 16
8
8
x=2
5𝑥 − 6𝑦 = −32
Find the solution set for 3𝑥 + 6𝑦 = 48
Procedures
Substitute the value of
the variable above into
either original equation
to solve for the
remaining unknown
variable.
What did that variable
equal?
𝐲=𝟕
𝟓𝒙 − 𝟔𝒚 = −𝟑𝟐
𝟑𝒙 + 𝟔𝒚 = 𝟒𝟖
Remember x = 2
3x + 6y = 48
3(2) + 6y = 48
6 + 6y = 48
-6
-6
6y = 42
6
6
y=7
5𝑥 − 6𝑦 = −32
Find the solution set for 3𝑥 + 6𝑦 = 48
Procedures
𝟓𝒙 − 𝟔𝒚 = −𝟑𝟐
𝟑𝒙 + 𝟔𝒚 = 𝟒𝟖
a) What is/are the solutions to the system? (2, 7)
b)If graphed, what type of lines would this
system form and how can you determine this
WITHOUT graphing the system?
Intersecting lines because there is one
solution.
c) Graph the system with your calculator to
verify the solution set is correct.
𝑥+𝑦 =5
Find the solution set for 3𝑥 − 𝑦 = 7
Procedures
Make sure both equation are in
Standard Form:
𝐀𝐱 + 𝐁𝐲 = 𝐂
(_____________)
While in standard form, which
terms has the same coefficient?
y and -y
Including the operation in front
of each term, what operation
would cancel out those terms?
𝐀𝐝𝐝𝐢𝐭𝐢𝐨𝐧
Use this operation to eliminate
the terms . Solve for the
remaining variable.
The remaining variable equal?
𝐱=𝟑
𝒙+𝒚=𝟓
𝟑𝒙 − 𝒚 = 𝟕
Both equations are already in
Standard From
x+y=5
+ 3x - y = 7
4x
= 12
4
4
x=3
𝑥+𝑦 =5
Find the solution set for 3𝑥 − 𝑦 = 7
Procedures
Substitute the value of
the variable above into
either original equation
to solve for the
remaining unknown
variable.
What did that variable
equal?
𝐲=𝟐
𝒙+𝒚=𝟓
𝟑𝒙 − 𝒚 = 𝟕
Remember x = 3
x+y=5
3+y=5
-3
-3
y=2
𝑥+𝑦 =5
Find the solution set for 3𝑥 − 𝑦 = 7
Procedures
𝒙+𝒚=𝟓
𝟑𝒙 − 𝒚 = 𝟕
a) What is/are the solutions to the system? (3, 2)
b)If graphed, what type of lines would this
system form and how can you determine this
WITHOUT graphing the system?
Intersecting lines because there is one
solution.
c) Graph the system with your calculator to
verify the solution set is correct.
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
𝟒𝒙 + 𝒚 = 𝟑
Procedures
Make sure both equations are in
Standard Form:
𝐀𝐱 + 𝐁𝐲 = 𝐂
(_____________)
Pick a variable to eliminate:
y
What is the least common
multiple of both coefficients of
those terms?
𝟐
Multiply each equation by a
number so the chosen variable to
eliminate can have the same
coefficients as the LCD
(you may only need to multiply
one equation)
−𝒙 − 𝟐𝒚 = 𝟖
Both equations are already in
Standard From
However, what is different about this
system when in Standard Form?
The terms do NOT have like coefficients.
2( 4x + y = 3 )
- x - 2y = 8
8x + 2y = 6
-x – 2y = 8
Decide what operation should now be
used to eliminate the y term and complete
8x + 2y = 6
+-x – 2y = 8
7x
= 14
7
7
x=2
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
Substitute the value of
the variable above into
either original equation
to solve for the
remaining unknown
variable.
What did that variable
equal?
𝐲 = −𝟓
𝟒𝒙 + 𝒚 = 𝟑
−𝒙 − 𝟐𝒚 = 𝟖
Remember x = 2
4x + y = 3
4(2) + y = 3
8+y=3
-8
-8
y = -5
4𝑥 + 𝑦 = 3
Find the solution set for −𝑥 − 2𝑦 = 8
Procedures
𝟒𝒙 + 𝒚 = 𝟑
−𝒙 − 𝟐𝒚 = 𝟖
a) What is/are the solutions to the system? (2, -5)
b)If graphed, what type of lines would this
system form and how can you determine this
WITHOUT graphing the system?
Intersecting lines because there is one
solution.
2𝑥 + 5𝑦 = −22
Find the solution set for 10𝑥 + 3𝑦 = 22
𝟐𝒙 + 𝟓𝒚 = −𝟐𝟐
Procedures
Make sure both equations are in
Standard Form:
𝐀𝐱 + 𝐁𝐲 = 𝐂
(_____________)
Pick a variable to eliminate:
x
𝟏𝟎𝒙 + 𝟑𝒚 = 𝟐𝟐
Both equations are already in
Standard From
Notice none of the terms are equal again.
5( 2x + 5y = -22 )
10x + 3y = 22
10x + 25y = -110
10 x + 3y = 22
What is the least common multiple
of both coefficients of those terms? Decide what operation should now be
used to eliminate the y term and complete
𝟏𝟎
Multiply each equation by a
number so the chosen variable to
eliminate can have the same
coefficients as the LCD
(you may only need to multiply
one equation)
10x + 25y = -110
10x + 3y = 22
22y = -132
22
22
y = -6
2𝑥 + 5𝑦 = −22
Find the solution set for 10𝑥 + 3𝑦 = 22
Procedures
Substitute the value of
the variable above into
either original equation
to solve for the
remaining unknown
variable.
What did that variable
equal?
x= 𝟒
𝟐𝒙 + 𝟓𝒚 = −𝟐𝟐
𝟏𝟎𝒙 + 𝟑𝒚 = 𝟐𝟐
Remember y = -6
2x + 5y = -22
2x + 5(-6) = -22
2x – 30 = -22
+30 +30
2x = 8
2
2
x=4
2𝑥 + 5𝑦 = −22
Find the solution set for 10𝑥 + 3𝑦 = 22
Procedures
𝟐𝒙 + 𝟓𝒚 = −𝟐𝟐
𝟏𝟎𝒙 + 𝟑𝒚 = 𝟐𝟐
a) What is/are the solutions to the system? (4, -6)
b)If graphed, what type of lines would this
system form and how can you determine this
WITHOUT graphing the system?
Intersecting lines because there is one
solution.
𝑦 = −𝑥 + 3
Find the solution set for 2𝑥 + 2𝑦 = 6
𝒚 = −𝒙 + 𝟑
Procedures
Make sure both equation are in
Standard Form:.
Pick a variable to eliminate:
x
What is the least common
multiple of both coefficients of
those terms?
𝟐
Multiply each equation by a
number so the chosen variable to
eliminate can have the same
coefficients as the LCD
(you may only need to multiply
one equation)
y = -x + 3
+x + x
x+y=3
2( x + y = 3)
2x + 2y = 6
𝟐𝒙 + 𝟐𝒚 = 𝟔
Already in
Standard Form
2x + 2y = 6
2x + 2y = 6
Decide what operation should now be
used to eliminate the y term and complete
2x + 2y = 6
2x + 2y = 6
0=0
𝑦 = −𝑥 + 3
Find the solution set for 2𝑥 + 2𝑦 = 6
Procedures
𝒚 = −𝒙 + 𝟑
𝟐𝒙 + 𝟐𝒚 = 𝟔
a) What is/are the solutions to the system?
There are infinite many solutions.
b)If graphed, what type of lines would this
system form and how can you determine this
WITHOUT graphing the system?
Coinciding lines because both sides of
the equation are equal.
−3𝑥 + 𝑦 = 1
Find the solution set for 3𝑥 − 𝑦 = 6
−𝟑𝒙 + 𝒚 = 𝟏
Procedures
Make sure both equation are in
Standard Form:.
Pick a variable to eliminate:
x
What operation would cancel out
those terms?
addition
Multiply each equation by a
number so the chosen variable to
eliminate can have the same
coefficients as the LCD
(you may only need to multiply
one equation)
𝟑𝒙 − 𝒚 = 𝟔
Already in
Standard Form
-3x + y = 1
+
3x – y = 6
0=7
−3𝑥 + 𝑦 = 1
Find the solution set for 3𝑥 − 𝑦 = 6
Procedures
−𝟑𝒙 + 𝒚 = 𝟏
𝟑𝒙 − 𝒚 = 𝟔
a) What is/are the solutions to the system?
There are no solutions.
b)If graphed, what type of lines would this
system form and how can you determine this
WITHOUT graphing the system?
Parallel lines because 0 can never
equal 7.