5 – 4:
Word Problems and a System of Two Equations
Selected Worked Homework Problems
1.
The sum of two positive numbers is 12. The difference of the two numbers is 6. Find both of the
numbers.
Sentence
Sentence
Sentence
The sum of two
positive numbers is 12
If we ADD the two numbers
x+y
the sum is (=) 12
x + y = 12
The difference of two
positive numbers is 6
If we SUBTRACT the two numbers
x–y
the difference is (=) 6
x –y=6
⎧ x + y = 12
add both equations
⎨
x
−
y
=
6
⎩
2x =18
solve for x
x =9
plug x =9 into either of the original
equations and solve for y
plug x =9 into x + y = 12
9 + y = 12
y=3
Answer : The numbers are 9 and 3
Check
9 + 3 = 12
9−3= 6
Math 100 !
Section 5 – 4 HW WKD
!
© 2016 Eitel
2.
The sum of two positive numbers is 25. The difference of the two numbers is 13. Find both of
the numbers.
Sentence
Sentence
Sentence
The sum of two
positive numbers is 25
If we ADD the two numbers
x+y
the sum is (=) 125
x + y = 25
The difference of two
positive numbers is 13
If we SUBTRACT the two numbers
x–y
the difference is (=) 13
x – y = 13
⎧ x + y = 25
add both equations
⎨
⎩ x − y = 13
2x = 38
solve for x
x =19
plug x =19 into either of the original
equations and solve for y
plug x =19 into x + y = 25
19 + y = 25
y=6
Answer : The numbers are 19 and 6
Check
19 + 6 = 25
19 − 6 = 13
Math 100 !
Section 5 – 4 HW WKD
!
© 2016 Eitel
3.
The sum of two positive numbers is 52. One number is 8 less than four times the other number.
Find both of the numbers.
Sentence
Sentence
Sentence
The sum of two
positive numbers is 52
If we ADD the two numbers
x+y
the sum is (=) 52
x + y = 52
One number (x) is =
x=
8 less than four times the other number (y)
4y – 8
x = 4y – 8
One number is
8 less than four times
the other number
⎧ x + y = 52
⎨
⎩ x = 4y − 8
substitutue the 4y − 8 for x in the top equation
(x) + y = 52
(4y − 8) + y = 52
substitutue the 3y − 12 for x
solve for y
4y − 8 + y = 52
5y − 8 = 52
5y = 60
y = 12
plug y =12 into the second of the original
equations and solve for x
x = 4y − 8 and y = 12
x = 4(12) − 8
x = 48 − 8
x = 40
Answer : The numbers are 12 and 40
Check
40 + 12 = 52
40 = 4(12) − 8
Math 100 !
Section 5 – 4 HW WKD
!
© 2016 Eitel
4.
The sum of two positive numbers is 65. One number is 5 more than three times the other
number. Find both of the numbers.
Sentence
Sentence
Sentence
The sum of two
positive numbers is 65
If we ADD the two numbers
x+y
the sum is (=) 65
x + y = 65
One number (x) is =
x=
5 more than three times the other number (y)
3y + 5
x = 3y + 5
One number is
5 more than three times
the other number
⎧ x + y = 65
⎨
⎩ x = 3y + 5
substitutue the 3y + 5 for x in the top equation
(x) + y = 65
substitutue the 3y − 12 for x
(3y + 5) + y = 65 solve for y
3y + 5 + y = 65
4y + 5 = 65
4y = 60
y = 15
plug y =15 into the second of the original
equations and solve for x
x = 3y + 5 and y = 15
x = 3(15) + 5
x = 45 + 5
x = 50
Answer : The numbers are 50 and 15
Check
50 + 15 = 65
50 = 3(15) + 5
Math 100 !
Section 5 – 4 HW WKD
!
© 2016 Eitel
7. The total number of doors and windows in a house is 28. The number of windows is six times the
number of doors. Find the total number of windows and doors.
Sentence
Sentence
Sentence
The total number of doors
and windows is 28
If we ADD the number of doors (D)
and windows (W) the sum is 28
D + W = 28
The number of windows is
six times
the number of doors
The number of windows (W) is =
W=
six times the number of doors
6D
W = 6D
⎧ D + W = 28
⎨
⎩W = 6D
substitutue the 6D for W in the top equation
D + (W ) = 28
D + (6D) = 28
substitutue 6D for W
solve for D
D + 6D = 28
7D = 28
D=4
plug D = 4 into the second equation and solve for W
W = 6D and D = 4
W = 6(4)
W = 24
ANSWER : There are 4 doors and 24 windows
Check
4 + 24 = 28
24 = 6(4)
Math 100 !
Section 5 – 4 HW WKD
!
© 2016 Eitel
8. The total number of nurses and doctors in a hospital is 42. The number of nurses is 6 more than
twice the number of doctors. Find the total number of doctors and nurses.
Sentence
Sentence
Sentence
The total number of nurses
and doctors is 42
If we ADD the number of nurses (N)
and doctors (D) the sum is 42
N + D = 42
The number of nurses is
6 more than six twice
the number of doctors
The number of nurses (N) is =
N=
six more that twice the number of doctors
2D + 6
N = 2D + 6
⎧ N + D = 42
⎨
⎩ N = 2D + 6
substitutue the 2D + 6 for N in the top equation
N + D = 42
D + 2D + 6 = 42
substitutue 2D + 6 for N
solve for D
3D + 6 = 42
3D = 36
D = 12
plug D = 12 into the second equation and solve for N
N
N
N
N
= 2D + 6 and D = 12
= 2(12) + 6
= 24 + 6
= 30
ANSWER : There are 12 doctors and 30 nurses
Check
12 + 30 = 42
30 = 2(12) + 6
Math 100 !
Section 5 – 4 HW WKD
!
© 2016 Eitel