Solving Systems of
Equations by
Substitution
ALGEBRA 1
SOL A4e
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
4𝑥 + 2𝑦 = 8
2x + y = 4
-2x
-2x
y = -2x + 4
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.