Real world problems
Bill, Lala, and Jane are each making a bouquet using roses, daisies, and lilacs. Bill purchased 8 flowers and Lala bought 19 flowers. She bought twice as many roses, twice as many daisies, and three times as many lilacs as Bill. Jane bought 20 flowers. She bought three times as many roses, twice as many daisies, and twice as many lilacs as Bill. How many of each type of flower did Bob purchase? (Three equations and can be solved using a system of equations).
Bill's equation: x + y + z = 8.
Lala's equation: 2x + 2y + 3z = 19.
Jane's equation: 3x + 2y + 2z = 20.
Solve the following system of equations using elimination: x+y =9
x−y=3
x=6
y=3
y = 3x − 7
y = −2x + 3
(2,-1) is the only solution to this system of equations.
y = 2x^2 − 7x + 9
3x + y = 15
The solutions to this system of equations are (−1, 18) and (3, 6).
x^3 − 4x^2 + x − 2 = 2x + 2.
y = x3 − 4x2 + x − 2
y = 2x + 2
Lila wants to buy a new phone, the Findy phone plan charges her $25 monthly with no fees. The Discovo phone plan only charges $10 monthly but has a $50 set-up fee. Which plan is better for Lila? Graph the system of equations to find out!
Findy: y=25x and Discovo: y=10x+50, The graphs intersect at 10/3 and after 10/3 months, the two companies will have the same cost.
2x+y=3
4x+2y=6
he resulting equation, 0x + 0y = 0, is the same as 0 = 0.
3x + y = −2
9x + 3y = 3
No Solutions
What do the two solutions to the system of equations represent in Nonlinear Equations?
These are the points that the graphs of the equations will intersect.
y=-2x^2+1
y=4x-5
The solutions to the quadratic and linear system of equations are the points (−3, −17) and (1, −1).
You've been working the box office at a movie theater for a year now and want a promotion. Your supervisor asks you to find out how many adults and how many children attended the recent screening of “Amazing Algebra 2.” The total number of adult and child tickets sold was 150. Each adult ticket costs 9 dollars and one child ticket costs 6 dollars. The movie theater made $1,155.
a + c = 150
9a + 6c = 1155
a=85 adult tickets sold and c=65 children tickets sold.
x+y=−1
3x+3y=6
The resulting equation, 0x + 0y = 9, is the same as 0 = 9.
−2x + y = 1
−6x + 3y = 3
Infinite Solutions
y = x^2 + 3x − 5
2x + y = −5
The solutions to this system are (0, −5) and (−5, 5).
y=x^3-4x^2+x+3
y=2x-1
The solutions to the cubic and linear system of equations are the points (−1, −3), (1, 1), and (4, 7).
You are designing a sign for a business to be displayed near a major road in Phoenix. You have enough wood to build a rectangular sign that has a perimeter of 18 feet. You have enough money in your budget to purchase enough material for a sign that is 18 ft2 in area. How can you plan the biggest rectangular sign possible? First, look at the constraints. The perimeter cannot be more than 18 ft, and the area cannot be more than 18 ft^2 . Using l as the variable for the length and w for the width, the two constraint equations would be 2l + 2w = 18 (the perimeter) and l•w = 18 (the area).
l = 9 − w
l = 18 over w
The sign can be built with a width of 3 feet and a length of 6 feet or with a width of 6 feet and a length of 3 feet.
x+2y−z =−3
2x−2y+2z=8
2x−y+3z=9
Since x=1, y=−1 and z=2, the solution is (1,−1,2). Graphically, this represents the only point where the three planes intersect.
y = 5x − 11
y = −2x + 3
The solution to the system of equations using f(x) and g(x) is (1, 0).
y^2 + x^2 = 20
y = −2x
There are two solutions to the system of equations, (2, −4) and (−2, 4).
2x^2-10=-x^3+4x^2-x+2
y=2x^2-10
y=-x^3+4x^2-x+2 The solution to the quadratic and cubic system of equations is the point (3, 8). Therefore, x = 3 is the solution to the original equation.
The height of a falling leaf is modeled by the equation h = −4.9t2 + 30 where h is the height in meters, and t is the time in seconds. A remote controlled helicopter is flying up to catch the leaf in an amazing display of piloting. The helicopter’s height is modeled by h = 2t. When will the helicopter catch the leaf?
h = −4.9t2 + 30
h = 2t
(−2.69, −5.37) unreasonable and (2.28, 4.56)
3x+2y−z=11
x−3y+2z= −1
2x−y−3z=12
Since x=3, y=0 and z=-2, the solution is (3,0,-2). Graphically, this represents the only point where the three planes intersect.
y=5x−3
y=−2x+4
The solution to the system of equations is (1,2)
y = x - 2, over x + 3
y = −x + 2
The solutions to the system of equations are (−4, 6) and (2, 0).
y^2+x^2=36
y=-3x+5
The solutions to the circle and linear system of equations are the points (−0.33, 5.99) and (3.33, −4.99).