Please Help!!! And explain!!! Solve for x 7(5x-4)-1=14-8x
1 answer:
You might be interested in
Answer:
The first customer that will get four free tickets is 50th customer
Step-by-step explanation:
Find the least common multiple of numbers 10 and 25. First, factorize these numbers:
When finding LCM, first write the all common multiples (underlined 5) and then multiply them by remaining multiples (2 and 5). You get 50 as LCM(10,25). This means that each 50th customer will get four free tickets.
Answer:
The absolute minimum of f(x,y) = 8.107
The absolute maximum of f(x,y) = 24.326
Step-by-step explanation:
f(x,y) = xy. The constraint equation is 2x² + 3y² - 3xy = 49
Let g(x,y) = 2x² + 3y² - 3xy - 49
df/dx = y and df/dy = x , dg/dx = 4x - 3y and dg/dy = 6y - 3x
Using Lagrange multipliers,
df/dx = λdg/dx and df/dy = λdg/dy
So,
y = λ(4x - 3y) (1 )and x = λ(6y - 3x) (2)
y = 4λx - 3λy
y + 3λy = 4λx
y(1 + 3λ) = 4λx
y = 4λx/(1 + 3λ)
Substituting y into (2), we have
x = λ(6y - 3x)
x = λ(6[4λx/(1 + 3λ)] - 3x)
x = 24λ²x/(1 + 3λ) - 3λx
24λ²x/(1 + 3λ) - 3λx - x = 0
[24λ²/(1 + 3λ) - 3λ - 1]x = 0
⇒ [24λ²/(1 + 3λ) - 3λ - 1] = 0 since x ≠ 0
[24λ²/(1 + 3λ) - 3λ - 1] = 0
⇒[24λ²/(1 + 3λ) - (3λ + 1)] = 0
[24λ² - (3λ + 1)²] = 0
24λ² - 9λ² - 6λ - 1 = 0
15λ² - 6λ - 1 = 0
Using the quadratic formula,
λ =
λ = (6 + 4√6)/30 or (6 - 4√6)/30
λ = (3 + 2√6)/15 = 0.527 or (3 - 2√6)/15 = -0.127
Substituting y into the constraint equation, we have
2x² + 3y² - 3xy = 49
2x² + 3(4λx/(1 + 3λ))² - 3x(4λx/(1 + 3λ)) = 49
2x² + 12λ²x²/(1 + 3λ))² - 12λx²/(1 + 3λ) = 49
[2 + 12λ²/(1 + 3λ)² - 12λ/(1 + 3λ)}x² = 49
[2(1 + 3λ)² + 12λ² - 12λ(1 + 3λ)]x²/(1 + 3λ)² = 49
[2(1 + 6λ + 9λ²) + 12λ² - 12λ + 36λ²)]x²/(1 + 3λ)² = 49
[2 + 12λ + 18λ² + 12λ² - 12λ + 36λ²)]x²/(1 + 3λ)² = 49
[2 + 6λ²]x²/(1 + 3λ)² = 49
x² = 49(1 + 3λ)²/(2 + 6λ²)
x² = 49(1 + 3λ)²/2(1 + 3λ²)
x = √[49(1 + 3λ)²/2(1 + 3λ²)]
x = ±7√[(1 + 3λ)²/2(1 + 3λ²)]
Substituting λ = (3 + 2√6)/15 = 0.527 or (3 - 2√6)/15 = -0.127
x = ±7√[(1 + 3(0.527))²/2(1 + 3(0.527)²)] or ±7√[(1 + 3(-0.127))²/2(1 + 3(-0.127)²)]
x = ±7√[(6.662/3.666] or ±7√[0.3831/1.9032)]
x = ±7√1.8172 or ±7√0.2012
x = ±9.436 or ±3.141
Substituting x and λ into y, we have
y = 4λx/(1 + 3λ)
y = 4(0.527)(±3.141)/(1 + 3(0.527)) or 4(-0.127)(±3.141)/(1 + 3(-0.127))
y = ±6.6621/2.581 or ±1.5956/0.619
y = ±2.581 or ±2.578
The minimum value of f(x,y) is gotten at the minimum values of x and y which are x = -3.141 and y = -2.581
So f(-3.141,-2.581) = -3.141 × -2.581 = 8.107
The maximum value of f(x,y) is gotten at the minimum values of x and y which are x = +9.436 and y = +2.578
So f(+9.436,+2.578) = +9.436 × +2.578 = 24.326