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Sauron [17]
3 years ago
8

What is the best first step for solving the given system using substitution while avoiding fractions -9x+4y=10 -9x+3y=3

Mathematics
2 answers:
Setler [38]3 years ago
8 0

Answer:

B is the answer

Step-by-step explanation:

Sever21 [200]3 years ago
7 0

Answer:

Solve for x in the first equation. ⇒ the first answer

Step-by-step explanation:

* Lets study how to solve the system of the equation using

 the substitution method

- The first equation is ax + by = c

- The second equation is dx + ey = f

* To chose the first step

- Solve for x or y depends on the integer answer

EX: ax = c - by ⇒ x = c/a - (b/a)y

      b/a = integer , c/a = integer ⇒ best choice

- If not solve for next variable

EX: by = c - ax ⇒ y = c/b - (a/b)y

      a/b = integer , c/b = integer ⇒ best choice

- If not repeat with the second equation to find the best choice

* Lets check our equations:

∵ -9x + 4y = 10 ⇒ (1)

∵ -9x + 3y = 3 ⇒ (2)

* If we start with the first equation and solve for x

∵ -9x = 10 - 4y ⇒ (2)

- It will be difficult if we divide by -9, but the second equation

 has -9x, so we will solve of -9x not for x and substitute the value

 of -9x in the second equation

* Substitute (3) in (2)

∴ 10 - 4y + 3y = 3 ⇒ collect the like terms

∴ -y = -7 ⇒ y = 7

substitute the value of in (3)

∴ -9x = 10 - 4(7) ⇒ -9x = -18 ⇒ ÷ -9 in both sides

∴ x = 3

∴ The best choice to avoid fraction is:

  Solve for x in the first equation.

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zysi [14]

Given <em>z</em> = 3 + <em>i</em>, right away we can find

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<em>z</em> ² = (3 + <em>i </em>)² = 3² + 6<em>i</em> + <em>i</em> ² = 9 + 6<em>i</em> - 1 = 8 + 6<em>i</em>

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(d) polar form

First find the argument:

arg(<em>z</em>) = arctan(1/3)

Then

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<em>z</em> = √10 exp(<em>i</em> arctan(1/3))

or

<em>z</em> = √10 (cos(arctan(1/3)) + <em>i</em> sin(arctan(1/3))

(c) square root

Any complex number has 2 square roots. Using the polar form from part (d), we have

√<em>z</em> = √(√10) exp(<em>i</em> arctan(1/3) / 2)

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√<em>z</em> = √(√10) exp(<em>i</em> (arctan(1/3) + 2<em>π</em>) / 2)

Then in standard rectangular form, we have

\sqrt z = \sqrt[4]{10} \left(\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) + i \sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right)\right)

and

\sqrt z = \sqrt[4]{10} \left(\cos\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right) + i \sin\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right)\right)

We can simplify this further. We know that <em>z</em> lies in the first quadrant, so

0 < arg(<em>z</em>) = arctan(1/3) < <em>π</em>/2

which means

0 < 1/2 arctan(1/3) < <em>π</em>/4

Then both cos(1/2 arctan(1/3)) and sin(1/2 arctan(1/3)) are positive. Using the half-angle identity, we then have

\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1+\cos\left(\arctan\left(\dfrac13\right)\right)}2}

\sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1-\cos\left(\arctan\left(\dfrac13\right)\right)}2}

and since cos(<em>x</em> + <em>π</em>) = -cos(<em>x</em>) and sin(<em>x</em> + <em>π</em>) = -sin(<em>x</em>),

\cos\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{1+\cos\left(\arctan\left(\dfrac13\right)\right)}2}

\sin\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{1-\cos\left(\arctan\left(\dfrac13\right)\right)}2}

Now, arctan(1/3) is an angle <em>y</em> such that tan(<em>y</em>) = 1/3. In a right triangle satisfying this relation, we would see that cos(<em>y</em>) = 3/√10 and sin(<em>y</em>) = 1/√10. Then

\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1+\dfrac3{\sqrt{10}}}2} = \sqrt{\dfrac{10+3\sqrt{10}}{20}}

\sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1-\dfrac3{\sqrt{10}}}2} = \sqrt{\dfrac{10-3\sqrt{10}}{20}}

\cos\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{10-3\sqrt{10}}{20}}

\sin\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{10-3\sqrt{10}}{20}}

So the two square roots of <em>z</em> are

\boxed{\sqrt z = \sqrt[4]{10} \left(\sqrt{\dfrac{10+3\sqrt{10}}{20}} + i \sqrt{\dfrac{10-3\sqrt{10}}{20}}\right)}

and

\boxed{\sqrt z = -\sqrt[4]{10} \left(\sqrt{\dfrac{10+3\sqrt{10}}{20}} + i \sqrt{\dfrac{10-3\sqrt{10}}{20}}\right)}

3 0
3 years ago
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Answer:

Hi there!

Your answer is:

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<h3>CloutAnswers</h3>
8 0
3 years ago
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