Applying Systems of
Equations – Part 1
Honors Math – Grade 8
1
Twice one number added to
another number is 18. Four times
the first number minus the other
number is 12. Find the numbers.
Let x represent the first number and y represent the second number. Translate
each sentence into an algebraic equation.
1. Write the equations in column form and add.
2 x  y  18
+ 4x  y  12
Define the Variables
Twice one # added to another is 18.
The y variable is
eliminated
because 1 + -1 = 0
Solve the equation
2. Now substitute x = 5 in either equation and solve.
The numbers are 5 and 8.
2x + y = 18
4 times the first minus the other is 12.
4x – y = 12
2
One number added to twice another
number is 13. Four times the first
number added to twice the other
number is -2. What are the numbers?
Let a represent the first number and b represent the second number. Translate
each sentence into an algebraic equation.
1. Write the equations in column form; subtract
a  2b  13
- 4a  2b  2
Define the Variables
One # added to twice another is 13.
The a variable is
eliminated
because 2 – 2 = 0
a + 2b = 13
4 times the first added to twice the
other is -2.
4a + 2b = -2
Solve the equation
2. Now substitute a = -5 in either equation and solve.
The numbers are -5 and 9.
3
Define the
Variables
Let a = the cost for an adult ticket
and s = the cost of a student ticket.
1. Write the equations in column form; subtract
2a  5s  77
- 2a  7s  95
The a variable is
eliminated
because 2 – 2 = 0
Solve the equation
2. Now substitute s = 9 in either equation and solve.
An adult ticket costs $16 and
a student ticket costs $9.
A youth group traveling in two vans visited
Mammoth Cave in Kentucky. The number
of people in each van and the total cost of a
tour of the cave are shown in the table.
Find the adult price and the student price of
the tour.
Van
Adults
Students
Total
Cost
A
2
5
$77
B
2
7
$95
Write a
system of
equations.
Adults + Students = TC
2a + 5s = 77
2a + 7s = 95
4
Define the
Variables
Let g = Rich Gannon’s earnings & w
= Charles Woodson’s earnings.
In 2003, Rich Gannon, the Oakland Raiders
quarterback, earned $4 million more than
Charles Woodson, the Raiders cornerback.
Together they cost the Raiders
approximately $9 million. How much did
each make?
Write a
system of
equations.
One equation is solved for g; Substitute g= w+4
gw9
w 4 w  9
Substitute w + 4
for g in the first
equation.
Group like terms
Solve.
2. Now substitute w = 2.5 in either equation and solve.
Together they cost the Raiders
9 million.
g+w=9
Rich Gannon earned 4 million
more than Woodson.
g=w+4
Rich Gannon
made $6.5
million and
Charles
Woodson
made $2.5
million.
g  w  9

g  w  4
5
Define the
Variables
Let y = Yankee wins and
= Reds wins.
r
One equation is solved for y; Substitute y = 5.2r
Substitute 5.2r for
y in the first
equation.
Group like terms
Solve.
2. Now substitute r = 5 in either equation and solve.
The Yankees won 26 World Series and the Reds
won 5 World Series.
The New York Yankees and the
Cincinnati Reds together have won a
total of 31 World Series. The Yankees
have won 5.2 times as many as the
Reds. How many Worlds Series did
each time win?
Write a
system of
equations.
Together they won a total of 31
World Series.
y + r = 31
The Yankees won 5.2 times as
many as the Reds
y = 5.2r
 y  r  31

 y  5.2r
6
Define the
Variables
Let x = Angle X and
y = Angle Y.
1. Write the equations in column form and add.
x  y  180
(+)  x  y  24
Angles X and Y are supplementary
and the difference between angle Y
and angle X is -24. Find the angle
measures.
Write a
system of
equations.
Supplementary angles are two
angles whose sum is 180.
x + y = 180
The difference between Angle
Y and Angle X is -24.
y – x = -24 or –x + y = -24
The x variable is
eliminated
because 1 + -1 = 0
Solve the equation
2. Now substitute y=78 in either equation and solve.
Angle X measures 102 degrees and Angle Y
measures 78 degrees.
 x  y  180

 y  x  24
7
Define the
Variables
Let b = the height of the building
and let g = the height of the statue.
1
1. Write the equations in column form and add.
b  g  326.6
(+) b  g  295.4
The total height of an office building and the
granite statue that stands on top of it is
326.6 feet. The difference in heights
between the building and the statue is 295.4
feet. How tall is the statue?
Write a
system of
equations.
The total height of the building
and the statue is 326.6
b + g = 326.6
The difference between them
is 295.4
b – g = 295.4
The g variable is
eliminated
because 1 + -1 = 0
Solve the equation
2. Now substitute b=311 in either equation and solve
The statue is 15.6 feet tall
b  g  326.6

b  g  295.4