Solving Linear Systems by
Substitution
Objective:
• Students will solve a linear system by
substitution method.
Use this method when ONE of the equation
has been solved for a variable.
Algebra Standards:
8 EE 8 Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in
two variables correspond to points of intersection of their graphs,
because points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables
algebraically, and estimate solutions by graphing the equations.
Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x +
2y = 6 have no solution because 3x + 2y cannot simultaneously be 5
and 6.
c. Solve real-world and mathematical problems leading to two linear
equations in two variables. For example, given coordinates for two
pairs of points, determine whether the line through the first pair of
points intersects the line through the second pair.
Solve the system by graphing
x=3
y = 2x - 1
Review…
Solution
(3, 5)
x=3
Review…
Solve for y when x = 3
So, when x = 3…
y = 5 based on
the rule ( y = 2x -1)
Solution (3, 5)
Substitute the 3
into the x
#1 Substitution
Solve the linear system
y = 4x
{
Check answer!
x + y = 10
Solution:
y = 4x
x + y = 10
x + 4x = 10
y = 4(2)
1
y = 8
5x
5
= 10
5
x = 2
Solution is (2, 8)
#2 Substitution
Solve the linear system
x = 3y
{
Check answer!
2x – y = 10
Solution:
x = 3y
2x – y = 10
2(3y) – y = 10
x = 3(2)
x = 6
6y – y = 10
1 5y = 10
5
5
y =2
Solution is (6, 2)
#3 Substitution
Solve the linear system
x + 3y = -9
{
Check answer!
2x – 5y = 26
Solution:
0
x + 3y = -9
-3y
x
2x – 5y = 26
-3y
= -3y – 9
x = -3(-4)– 9
x = 12 – 9
x = 3
Solution is (3, -4)
2(-3y – 9) – 5y = 26
-6y – 18 – 5y = 26
0
-11y– 18 = 26
+18
1
+18
-11y = 44
-11
-11
y = -4
#4 Substitution
Solve the linear system
-x + y = 1
{
Check answer!
2x + y = -2
Solution:
0
-x + y = 1
+x
+x
y = x +1
y
y
= -1 + 1
= 0
2x + y = -2
2x + x + 1 = -2
0
3x + 1 = -2
-1
-1
1
Solution is (-1, 0)
3x = -3
3
3
x = -1
#5b Substitution- Word Problem
In one day the National Civil Rights Museum in Memphis,
Tennessee, admitted 321 adults and children and collected
$1590. The price of admission is $6 for an adult and $4 for a
child. How many adults and how many children were admitted
to the museum that day?
Solution:
Money
How many
x = adults
0
y = children
6x + 4y = 1590
x + y = 321
-y
x
-y
= -y + 321
x = -168 + 321
153 adults
x = 153
168 children
6(-y + 321) + 4y = 1590
-6y + 1926
+ 4y = 1590
0
-2y + 1926 = 1590
-1926
-1926
-2y = -336
y = 168
#6 Substitution
The length of a rectangle is 5 centimeters more than three times
the width. If the perimeter of the rectangle is 34 centimeters,
P = L + L + w +w
what are its dimensions?
P = 2L + 2w
L = 3w + 5
{
34 = 2L + 2w
Solution:
L = 3w + 5
L = 3(3) + 5
L =9 +5
L = 14
Check answer!
2L + 2w = 34
2(3w + 5) + 2w = 34
6w + 10 + 2w = 34
0
8w + 10 = 34
-10 -10
1
w = 3 8w = 24
Length is 14 cm and the width is 3cm
8
8
#7 Substitution
The sum of the ages of Petra and her mother is 54. Her
mother is 9 years more than twice as old as Petra. How old
are Petra and her mother
Check answer!
Solution:
x = Petra’s age
x + y = 54
y = 2x + 9
y = mother’s age
x + 2x + 9 = 54
3x + 9 = 54
-9 -9
1
3x = 45
3
3
x
= 15
y = 2(15) + 9
y = 30 + 9
y = 39
Petra is 15 years old.
Her mother is 39 years old.
Assignment
Book Pg. 284
# 5, 11, 13, 14, 17