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dlinn [17]
3 years ago
8

Solve the system y = -x + 7 and y = -0.5(x - 3)^2 + 8.

Mathematics
1 answer:
AlladinOne [14]3 years ago
8 0

Answer:

(1,6) & (7,0)

Step-by-step explanation:

y = -x + 7

y = -0.5(x - 3)² + 8

To solve the system, solve these two equations simultaneously

-x + 7 = -0.5(x - 3)² + 8

-x + 7 = -0.5(x² - 6x + 9) + 8

-x + 7 = -0.5x² + 3x - 4.5 + 8

0.5x² - 4x + 3.5 = 0

x² - 8x + 7 = 0

x² - 7x - x + 7 = 0

x(x - 7) - (x - 7) = 0

(x - 1)(x - 7) = 0

x = 1, 7

y = -1 + 7 = 6

y = -7 + 7 = 0

(1,6) (7,0)

Since the system has two distinct solutions, the line and the curve meet at two distinct poibts9: (1,6) & (7,0)

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Solve these recurrence relations together with the initial conditions given. a) an= an−1+6an−2 for n ≥ 2, a0= 3, a1= 6 b) an= 7a
8_murik_8 [283]

Answer:

  • a) 3/5·((-2)^n + 4·3^n)
  • b) 3·2^n - 5^n
  • c) 3·2^n + 4^n
  • d) 4 - 3 n
  • e) 2 + 3·(-1)^n
  • f) (-3)^n·(3 - 2n)
  • g) ((-2 - √19)^n·(-6 + √19) + (-2 + √19)^n·(6 + √19))/√19

Step-by-step explanation:

These homogeneous recurrence relations of degree 2 have one of two solutions. Problems a, b, c, e, g have one solution; problems d and f have a slightly different solution. The solution method is similar, up to a point.

If there is a solution of the form a[n]=r^n, then it will satisfy ...

  r^n=c_1\cdot r^{n-1}+c_2\cdot r^{n-2}

Rearranging and dividing by r^{n-2}, we get the quadratic ...

  r^2-c_1r-c_2=0

The quadratic formula tells us values of r that satisfy this are ...

  r=\dfrac{c_1\pm\sqrt{c_1^2+4c_2}}{2}

We can call these values of r by the names r₁ and r₂.

Then, for some coefficients p and q, the solution to the recurrence relation is ...

  a[n]=pr_1^n+qr_2^n

We can find p and q by solving the initial condition equations:

\left[\begin{array}{cc}1&1\\r_1&r_2\end{array}\right] \left[\begin{array}{c}p\\q\end{array}\right] =\left[\begin{array}{c}a[0]\\a[1]\end{array}\right]

These have the solution ...

p=\dfrac{a[0]r_2-a[1]}{r_2-r_1}\\\\q=\dfrac{a[1]-a[0]r_1}{r_2-r_1}

_____

Using these formulas on the first recurrence relation, we get ...

a)

c_1=1,\ c_2=6,\ a[0]=3,\ a[1]=6\\\\r_1=\dfrac{1+\sqrt{1^2+4\cdot 6}}{2}=3,\ r_2=\dfrac{1-\sqrt{1^2+4\cdot 6}}{2}=-2\\\\p=\dfrac{3(-2)-6}{-5}=\dfrac{12}{5},\ q=\dfrac{6-3(3)}{-5}=\dfrac{3}{5}\\\\a[n]=\dfrac{3}{5}(-2)^n+\dfrac{12}{5}3^n

__

The rest of (b), (c), (e), (g) are solved in exactly the same way. A spreadsheet or graphing calculator can ease the process of finding the roots and coefficients for the given recurrence constants. (It's a matter of plugging in the numbers and doing the arithmetic.)

_____

For problems (d) and (f), the quadratic has one root with multiplicity 2. So, the formulas for p and q don't work and we must do something different. The generic solution in this case is ...

  a[n]=(p+qn)r^n

The initial condition equations are now ...

\left[\begin{array}{cc}1&0\\r&r\end{array}\right] \left[\begin{array}{c}p\\q\end{array}\right] =\left[\begin{array}{c}a[0]\\a[1]\end{array}\right]

and the solutions for p and q are ...

p=a[0]\\\\q=\dfrac{a[1]-a[0]r}{r}

__

Using these formulas on problem (d), we get ...

d)

c_1=2,\ c_2=-1,\ a[0]=4,\ a[1]=1\\\\r=\dfrac{2+\sqrt{2^2+4(-1)}}{2}=1\\\\p=4,\ q=\dfrac{1-4(1)}{1}=-3\\\\a[n]=4-3n

__

And for problem (f), we get ...

f)

c_1=-6,\ c_2=-9,\ a[0]=3,\ a[1]=-3\\\\r=\dfrac{-6+\sqrt{6^2+4(-9)}}{2}=-3\\\\p=3,\ q=\dfrac{-3-3(-3)}{-3}=-2\\\\a[n]=(3-2n)(-3)^n

_____

<em>Comment on problem g</em>

Yes, the bases of the exponential terms are conjugate irrational numbers. When the terms are evaluated, they do resolve to rational numbers.

6 0
2 years ago
Is (-4, 2) a solution of 4x + 5y &lt; -7?
cupoosta [38]

Answer:

No

Step-by-step explanation:

Substitute the coordinates of the point into the left side of the inequality, evaluate and compare.

4(- 4) + 5(2) = - 16 + 10 = - 6 > - 7

Thus (- 4, 2 ) is not a solution

4 0
3 years ago
Simplify the expression show all your work 18 / 2 x 3 over 5 - 2
11Alexandr11 [23.1K]

Answer:

18÷2 × 3 / 5-2

9 ×3 / 3

27/3

9

tyvm this was my 100th answer :)

6 0
3 years ago
What is an estimation of 492.6 divided by 48
denpristay [2]
492.6/48 ~ To simply view this:

480/48 = 10
12.6/48 = 0.25 (roughly).
10.25 would roughly be your answer.

Though...
The actual answer is that:
12.6/48 = 0.2625
So... the answer: 10.2625 would be the answer.
Depending on where you have to estimate to:
10.2625
10.263
10.26
10.3

I hope that helps, have a great of your day! ^ ^
{-Ghostgate-}
4 0
3 years ago
Read 2 more answers
If h(x) is the inverse of f(x). what is the value of h(f(x))?
attashe74 [19]

Option C is correct

The value of h(f(x)) is, x

Inverse function:

An inverse function that undergoes the other function.

For example: if f(x) is the inverse of g(x) then;

for every x.

As per the statement:

If h(x) is the inverse of f(x).

by definition of inverse:

Therefore,  the value of h(f(x)) is, x

6 0
3 years ago
Read 2 more answers
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