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2 years ago

11

What values for a and b make the system inconsistent? What values for a and b make the system consistent and dependent? Explain.

1 answer:

emmainna [20.7K]2 years ago

8 0

can you add the selections ( a and b ) ?

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andrey2020 [161]

First observe that if $a+b>0$ ,

$(a + b)^2 = a^2 + 2ab + b^2 \\\\ \implies a + b = \sqrt{a^2 + 2ab + b^2} = \sqrt{a^2 + ab + b(a + b)} \\\\ \implies a + b = \sqrt{a^2 + ab + b \sqrt{a^2 + ab + b(a+b)}} \\\\ \implies a + b = \sqrt{a^2 + ab + b \sqrt{a^2 + ab + b \sqrt{a^2 + ab + b(a+b)}}} \\\\ \implies a + b = \sqrt{a^2 + ab + b \sqrt{a^2 + ab + b \sqrt{a^2 + ab + b \sqrt{\cdots}}}}$

Let $a=0$ and $b=x$ . It follows that

$a+b = x = \sqrt{x \sqrt{x \sqrt{x \sqrt{\cdots}}}}$

Now let $b=1$ , so $a^2+a=4x$ . Solving for $a$ ,

$a^2 + a - 4x = 0 \implies a = \dfrac{-1 + \sqrt{1+16x}}2$

which means

$a+b = \dfrac{1 + \sqrt{1+16x}}2 = \sqrt{4x + \sqrt{4x + \sqrt{4x + \sqrt{\cdots}}}}$

Now solve for $x$ .

$x = \dfrac{1 + \sqrt{1 + 16x}}2 \\\\ 2x = 1 + \sqrt{1 + 16x} \\\\ 2x - 1 = \sqrt{1 + 16x} \\\\ (2x-1)^2 = \left(\sqrt{1 + 16x}\right)^2$

(note that we assume $2x-1\ge0$ )

$4x^2 - 4x + 1 = 1 + 16x \\\\ 4x^2 - 20x = 0 \\\\ 4x (x - 5) = 0 \\\\ 4x = 0 \text{ or } x - 5 = 0 \\\\ \implies x = 0 \text{ or } \boxed{x = 5}$

(we omit $x=0$ since $2\cdot0-1=-1\ge0$ is not true)

3 0

1 year ago

X-2y=0 ,x²-y²=3<br>by step

masya89 [10]

Answer:

x=2, y=1

x=-2, y = -1

Step-by-step explanation:

x - 2y = 0

x = 2y

x² - y² = 3

(2y)² - y² = 3

3y² = 3

y² = 1

y = 1 or y = - 1

x = 2 or x = -2

5 0

3 years ago

A Ferris vil can accommodate 55 people in 15 minute how many people could ride the Ferris ville in 2 hours what is the rate per

AnnyKZ [126]

15 minutes is one quarter of an hour, so the number of people accommodated in an hour would be 4 x 55 = 220 people per hour

3 0

2 years ago

Use substitution to solve the system of equations :

mote1985 [20]

Answer:

x=1

y=-4

Step-by-step explanation:

4x+y=0 -> y= -4x

2(-4x)+x=-7

-8x+ x=-7

-7x=-7

x=-7/-7=1

y=-4 . 1 = -4

3 0

2 years ago

What are the coordinates of the hole in the graph of the function?

MakcuM [25]

Answer:

The hole is at (5,7)

Step-by-step explanation:

x^2 − 3x − 10

-------------------

x−5

Factor the numerator

(x-5)(x+2)

-------------------

x−5

There is a hole at x=5 since it will cancel in the numerator and the denominator

f(x) = x+2 and letting x = 5

f(5) = 5+2

The hole is at (5,7)

4 0

3 years ago