Solving Systems of Linear Equations and Problem Solving

Use the addition method to solve the system
{X+Y = 83
{X-Y = 17
X+X=83
X-Y=17
2X = 100

Isolate X

2X/2 = 100/2
X = 50
Use 50 in place of X

50 + Y = 83 Subtract 50 from both sides
-50 = -50

Y = 33

Ann has been pricing train fares for group rates. One group consists of 3 adults and 4 children which costs \$159. The other group consists of 2 adults and 3 children which costs \$112. What are the prices of the Adult tickets and the child tickets. a = price of adult c = price of child
group 1 {3a + 4c = \$159
group 2 {2a + 3c = \$112

{ 9a + 12c = 477
{-8a – 12c = -448
a = 29

3(29)+4c = 159
87 +4c = 159
-87 -87
4c = 72
4c/4 = 72/4
c=18

A trail is 1200 miles and it goes through 4 towns. A group of hikers start at each town and walk toward each other. They meet after a time of 240 hours. If one group travels 1/2 mile per hour slower then the other group, find the rate of each group. r . t = d one group is X, one group is Y
240x + 240y = 1200
x + 1/2 = y
240x 240(x + 1/2) = 1200
240x + 240x + 120 = 1200
480x + 120 = 1200
– 120 = -120
480x = 1080
480x/480 = 1080/480 x = 2 1/4 mph
2 1/4 + 1/2 = y
2 3/4 = y
A chemist needs to prepare 12 liters of a 9% solution. Find the amount of a 4% solution and the amount of a 12% solution that she should mix to get the solution. group 1 {X + Y = 12
group 2 {0.04X + 0.12Y = 0.09(12)

-4X – 4Y = -48
4X + 12Y = 108
8Y = 60
8Y/8 = 60/8
Y = 7 1/2 Liters
4X + 12(7 1/2) = 108
4X + 90 = 108
-90 = -90
4X = 18
4X/4 = 18/4
X=4 1/2 Liters

Recall two angels are complimentary if the sum of their measures is 90 degrees. Find the measures of two complimentary angels if one angel is twice the other. group 1 {X + Y = 90
group 2 {Y = 2X

Use Y value of 2X in group 1
X + 2X = 90
3X = 90
3X/3 = 90/3
X = 30 degrees
Use X value of 30 in group 2
Y = 2(30)
Y = 60 degrees

New computer desks for a company estimate total cost of new desks to be \$6000. The cost of each desk is \$200. The revenue from each desk will be \$450. R(X) = 450X
C(X) = 6000 + 200X

450X = 6000 + 200X
-200X = – 200X
250X = 6000
250X/250 = 6000/250
X = 24 computer desks to break even

Revenue = 450(24) = \$10800
Cost = 6000 + 200(24)
6000 + 4800 = \$10800

The measure of the largest angle of a triangle is 80 degrees more than the measure of the smallest angle, and the measure of the remaining angle is 10 degrees more than the measure of the smallest angle. Find the measure of each angle. Sum of the degree of a triangle is 180 degrees
{Y = X + 80
{Z = X + 10
{X + Y + Z = 180
X + (X + 80) + (X + 10) = 180
3X + 90 = 180
-90 = -90
3X = 90
3X/3 = 90/3
X = 30
Y = 30+80=110
Y = 110
Z = 30 + 10
Z = 40

(30,110,40)