Linear Systems
Mary bought a total of 30 pieces of pool noodles and floaties for her pool party. The pool noodles each cost $5, and the floaties cost $24 each. She paid a total of $302.
Define the variables and equations needed to solve this system.
Let x represent the no. of pool noodles
Let y represent the no. floaties
Equation 1: x + y = 35
Equation 2: 5x + 24y = 302
Simplify:
(2x2+y3)0
Anything raised to the power of zero is always equivalent to 1.
What is the vertex form of a quadratic equation?
y = a(x - h)2 + k
Where (h, k) is the vertex.
What are the coordinates of the vertex of the following equation?
y = -5 (x - 2)2 -18
The vertex is (2,-18).
How can you differentiate the hypotenuse, adjacent, and opposite sides of a triangle?
Hypotenuse: The longest side on a triangle
Adjacent: It is next to the given angle
Opposite: It is opposite to the given angle
What is the solution to the system of equations using substitution?
2x + 3y = 7
x - 2y = 1
Solve for x:
x = 2y + 1
Substitute this value of x into the first equation:
2(2y + 1) + 3y = 7
4y + 2 + 3y = 7
7y + 2 = 7
7y = 5
y = 5/7
Substitute the value of y back into the second equation to solve for x:
x - 2 (5/7) = 1
x - 10/7 = 1
x = 10/7 + 1
x = 17/7
Therefore, the solution is (17/7,5/7)
Simplify the following quadratic expression:
x2 + 4x + 3
To simplify, we need to factor it.
Look for two numbers that will give the sum of the middle term (4) and the product of the constant (3).
? + ? = 4
? x ? = 3
In this case, the numbers are 1 and 3
Therefore, (x + 1)(x + 3)
Find the axis of symmetry if the x-intercepts are the following:
x-ints are: (-12,0) and (8,0)
We will use this fomula to find the axis of symmetry
x = r + s/2
r = -12 s = 8
x = -12 + 8/2
x = -2
Therefore, the axis of symmetry is at (-2,0).
State the quadratic formula.
x = (-b ± √(b^2 - 4ac)) / (2a)
Find the value of sin (45°).
sin(45°) = 0.707 or √2/2
Solve the following linear system for x and y using elimination:
4x + y = 8
2x - 3y = 1
Multiply the first equation by 3 and the second equation by 1 to make the coefficients of y equal:
12x + 3y = 24
2x - 3y = 1
Add both equations together to eliminate y:
14x = 25
x = 25/14
Substitute x into the first equation to solve for y:
4(25/14) + y = 8
25/7 + y = 8
y = - 25/7 + 8
y = 31/7
Therefore, the solution is (25/14,31/7)
Expand and simplify:
2(3x - 4)2 - (x-5)(x+5)
2(3x - 4)2 - (x-5)(x+5)
Expand by applying distributive property
2(9x2 - 24x + 16) - (x2 - 25)
18x2 - 48x + 32 - x2 +25
Combine like terms
17x2 - 48x + 57
What is the equation for a parabola that is reflected across the x-axis, is compressed by a factor of 1/2, is shifted 7 units to the left and down 3 units.
a= -1/2 h= -7 k= -3
Therefore, the equation is, y=-1/2 (x + 7)2 -3
Solve the quadratic equation using the quadratic formula: x2 - 5x - 14 = 0.
Using the quadratic formula, substitute a = 1, b = -5, and c = -14, and we get:
x = (-(-5) ± √((-5)^2 - 4(1)(-14))) / (2(1))
x = (5 ± √(25 + 56)) / 2
x = (5 ± √81) / 2
x = (5 ± 9) / 2
x = 7 or x = -2
Therefore, the solutions to the quadratic equation are x = 7 or x = -2.
Find the missing side length of the right triangle, given that sin(θ) = 0.6 and the opposite side is 8 m. Find the hypotenuse.
sin(θ) = opposite/hypotenuse:
0.6 = 8/x
x = 8/0.6
x ≈ 13.33
Therefore, the hypotenuse is approximately 13.33 m long.
Determine whether the equations has no solution, one solution, or infinite solutions:
5x - 2y = 3
10x - 4y = 6
Firstly, to compare the slopes, we can rewrite the equations in slope-intercept form (y = mx + b):
5x - 2y = 3
-2y = -5x + 3
-2/-2y = (-5 / -2)x + 3/-2
y = (5/2)x - 3/2
10x - 4y = 6
-4y = -10x + 6
-4/-4y = (-10/-4)x + 6 / -4
y = (10/4)x - 3/2
Both slopes are 5/2, and both y-intercepts are -3/2.
Therefore, the system has infinitely many solutions.
Find 5 possible values of b so that x2 + bx -24 can be factored.
? x ? = -24
? + ? = b
-4 x 6 = -24
-4 + 6 = 2
4 x -6 = -24
4 + -6 = -2
-12 x 2 = -24
-12 + 2 = -10
12 x -2 = -24
12 + -2 = 10
-8 x 3 = -24
-8 + 3 = -5
Therefore, the 5 possible values of b are, -2, 2, 20, -20 , and -5.
List all the transformations of the equation:
y = -2 (x + 2)2 -3
-It is reflected across the x axis
-Stretched by a factor of 2
-Shifted 2 units to the left
-Shifted 3 units down
The product of two consecutive integers is 182. Find the two integers.
Let x represent the smaller integer. Therefore, the larger integer is x + 1.
x(x + 1) = 182
Expand and rearrange the equation.
x2 + x - 182 = 0
Factor.
? + ? = 1
? x ? = -182
We get:
(x - 13)(x + 14) = 0
The two possible solutions:
x - 13 = 0 or x + 14 = 0
x = 13 or x = -14
Since we are looking for two consecutive integers, the answer is 13 and 14.
An observer is standing 200 meters away from a tall building. If the angle of elevation from the observer's eye to the top of the building is 30°, find the height of the building.
Let h represent the height of the building. the
tan(θ) = opposite/adjacent:
tan (30°) = h/200
h = 200 tan (30°)
h = 115.47
Therefore, the building is 115.47 m tall.
A candy shop sells two types of candies: chocolate bars and lollipops. The total revenue from selling 8 chocolate bars and 10 lollipops is $70. If a chocolate bar costs $4 and a lollipop costs $3, how many chocolate bars and lollipops did the candy shop sell?
Let x be the no. of Chocolate bars
Let y be the no. of lollipops
x + y = 18
4x + 3y = 70
Use Subtitution
x = 18 - y
4(18 - y) + 3y = 70
72 - 4y + 3y = 70
-y = -2
y = 2
Substitute the value of y back into Equation 1 to solve for x:
x + 2 = 18
x = 18 - 2
x = 16
Therefore, the candy shop sold 16 chocolate bars and 2 lollipops.
The path of a soccer ball kicked from a height of 2 meters can be modeled by the equation y = -5x2 + 30x + 2, where x represents the distance the ball has traveled and y represents its height. Determine the maximum height the ball reaches.
Find the vertex of the parabola.
x = -b/2a
a = -5 and b = 30
x = -30 / (2 * -5)
x = 3
Substitute x = 3 into the equation.
y = -5(3)2 + 30(3) + 2
y = -45 + 90 + 2
y = 47
Therefore, the maximum height reached by the soccer ball is 47 meters.
A store sells new laptops for $900 a piece, and they sell about 50 laptops each day. If the store decreases the price of the tablet by $80, then estimate that they will sell 8 more a day. What price should the store charge so they can maximize their revenue?
R = (Price)(h)
Factored Form to Standard
R = (900 - 80x) (50 + 8x)
R = 45000 + 7200x - 4000x - 640x2
R = - 640x2 + 3200x + 45000
Standard Form to Vertex Form
R = - 640x2 + 3200x + 45000
R = - 640 ( x2 - 5x +5 -5) + 45000
R = - 640 ( x - 5 )2 + 52200
Therefore, the vertex is (5,49000)
Calculate max R
x = 5
Rmax = (900 - 80x)
= (900 - 80(5))
= (900 - 400)
= $500.00
Therefore, to maximize thier revenue, they should proceed to do a sale and charge $500 per laptop.
A plane is flying at an altitude of 3100 meters. The angle of depression from the plane to a target on the ground is 12°. Find the horizontal distance from the plane to the target.
Let d represent the horizontal distance from the plane to the target.
tan (θ) = opposite/adjacent:
tan (12°) = 3100/d
d = 3100/tan (12°)
d = 14573.12
Therefore, the horizontal distance from the plane to the target is 14573.12 meters.