Download 9.2 Solve Linear Systems - Application Problems Number, Age, and

9.2 Solve Linear Systems - Application Problems
Number, Age, and Perimeter
• Remember, many real-life problems can be solved using a
system of linear equations.
Steps
Step 1: Identify the two unknown quantities.
Step 2: Define your two unknown quantities with variables.
Step 3: Represent two relationships using the variables by
writing two separate equations.
Step 4: Solve the system using your preferred method.
(Graphing, Substitution, or Elimination)
Step 5: Check your solution.
Ex 1:
The sum of two numbers is 100. Twice the first number
plus three times the second number is 275. What are the
two numbers?
Define the variables.
x = 1st number
y = 2nd number
Write the two equations
as a system.
x + y = 100
2x + 3y = 275
Write your final answer using
words and units, if necessary.
Solve the system.
x + y = 100
Multiply both sides by (-2).
2x + 3y = 275
-2x – 2y = -200
2x + 3y = 275 Eliminate the x’s.
y = 75
x + y = 100
x + 75 = 100
x = 25
Substitute for y and solve for x.
2(25) + 3(75) = 275 Check your solution.
50 + 225 = 275
275 = 275 Yes
1st number is 25 and 2nd number is 75.
Ex 2:
The smaller of two numbers is 3 less than the greater. If the greater
number is decreased by twice the smaller, the result is -5. Find the two
numbers.
Define the variables.
x = smaller number
y = larger number
Solve the system.
x=y-3
y – 2x = -5
Substitute for x with (y – 3).
Write the two equations
as a system.
x=y–3
y – 2x = -5
y – 2(y – 3) = -5 Solve for y.
y – 2y + 6 = -5
-y + 6 = -5
-y = -11
y = 11
x=y–3
x = 11 - 3
x=8
Write your final answer using
words and units, if necessary.
Substitute for y and solve for x.
11 – 2(8) = -5
Check your solution.
11 – 16 = -5
-5 = -5 Yes
Smaller number is 8 and larger number is 11.
Ex 3:
The difference in ages of 2 girls is 1 year. The sum of
their ages is 27 years. What are their ages?
Define the variables.
x = one girl’s age
y = the other girl’s age
Write the two equations
as a system.
x–y=1
x + y = 27
Solve the system.
x–y=1
x + y = 27
2x = 28
x = 14
Eliminate the y’s.
Solve for x.
x–y=1
Substitute for x and solve for y.
14 – y = 1
-y = -13
y = 13
x + y = 27
Check your solution.
14 + 13 = 27
27 = 27 Yes
Write your final answer using
words and units, if necessary.
One girl is age 14 and the other girl is age 13.
Ex 4:
George is 1 year less than 3 times as old as Neil. The sum
of their ages is 115. How old are Neil and George?
Define the variables.
g = George’s age
n = Neil’s age
Solve the system.
g = 3n - 1
g + n = 115
Substitute for g with (3n – 1).
Write the two equations
as a system.
g = 3n - 1
g + n = 115
(3n – 1) + n = 115
4n – 1 = 115
4n = 116
n = 29
Write your final answer using
words and units, if necessary.
Solve for n.
g = 3(29) – 1
g = 87 - 1
g = 86
Substitute for n, solve for g.
x + y = 115
86 + 29 = 115
115 = 115
Check your solution.
Yes
George’s is 86 years old. Neil is 29 years old.
Ex 5:
The length of a rectangle is 4 meters more than the width. The
perimeter of the rectangle is 40 meters. What do the length and the
width each measure? Note: For Perimeter problems, the formula
(P = 2ℓ + 2w) is often used as one of the equations of the system.
Define the variables.
ℓ = length in meters
w = width in meters
Solve the system.
ℓ=w+4
Substitute for ℓ with (w + 4).
2ℓ + 2w = 40
Write the two equations
as a system.
ℓ=w+4
2ℓ + 2w = 40
2(w + 4) + 2w = 40
2w + 8 + 2w = 40
4w + 8 = 40
4w = 32
w=8
ℓ=w+4
ℓ=8+4
ℓ = 12
Substitute for w, solve for ℓ.
2ℓ + 2w = 40
2(12) + 2(8) = 40
24 + 16 = 40
Write your final answer using
words and units, if necessary.
Solve for w.
Check your solution.
Yes
Length is 12 meters. Width is 8 meters.
Ex 6:
The difference between the length and width of a
rectangle is 7 centimeters. The perimeter of the
rectangle is 50 centimeters. Find the length and width.
Define the variables.
ℓ = length in centimeters
w = width in centimeters
Solve the system.
ℓ-w=7
ℓ=w+7
Substitute for ℓ with (w + 7).
2ℓ + 2w = 50
Write the two equations
as a system.
ℓ-w=7
2ℓ + 2w = 50
2(w + 7) + 2w = 50
2w + 14 + 2w = 50
4w + 14 = 50
4w = 36
w=9
ℓ-w=7
ℓ-9=7
ℓ = 16
Substitute for w, solve for ℓ.
2ℓ + 2w = 50
2(16) + 2(9) = 50
32 + 18 = 50
Write your final answer using
words and units, if necessary.
Solve for w.
Check your solution.
Yes
Length is 16 centimeters. Width is 9 centimeters.