Solutions To Simultaneous Equations (Review of Algebra II Conclusion)

 1) Solve the system below by substitution:

3x - y = 4

5x + 2y = 3

Solution:

Re- arrange top eqn.: y = 3x – 4

Substitute for y in 2nd: 5x + 2 (3x -4) = 3

Þ

5x + 6x – 8 = 3

11x =   11

x = 1

5(1) + 2y = 3

2y = 3 – 5 = -2  

y = -2/2 = -1

x= 1,  y  = -1 

2)Solve the linear system below graphically and by determinants:

2x - y = 10

5x + 2y = 7

Graphically:

By inspection: x = 3,   y = - 4

By Determinants: For main (D)

a1 =2, b1 = -1, c1 = 10

a2 = 5, b2 = 2, c2 = 7

So:

=  2(2) - 5(-1) = 4 + 5 = 9

Then x =  


=  10(2) - 7(-1)/  2(2) - 2(-1) = 27/9 =  3

Then y =  

 =   2(7) - 5(10) / 2 (2)- 5- (-1) = -36/ 9 = -4

x = 3, y = -4 

3) Solve the system below by substitution and by determinants:

4x - y = 5

3x + 2y = 12

By substitution:

Rewrite top eqn.   y =  4x  - 5

Substitute for y in 2nd:

3x + 2(4x - 5) = 12

3x +  8x - 10 = 12

11x = 10 + 12 = 22

x = 22/ 11 = 2

Solve for y using x:

3(2) + 2y  = 12

6 +  2y = 12

2y =  12 - 6 = 6

y = 6/2 = 3

 x = 2, y = 3

By Determinants: For main (D):

a1 =4, b1 = -1, c1 = 5

a2 = 3, b2 = 2, c2 = 12

So:

Then: D = (4) (2) - (3)(-1) = 8 +3 = 11

Then:

Substituting: x = 

=  (5)(2)) - (12)(-1) / 11

=  10 + 12/ 11  =  22/ 11  = 2

And:

Substituting:  y =

=  (4)(12)) - (3)(5) / 11

= (48 - 15)/ 11 = 33/ 11  = 3

x = 2, y = 3  

4) Solve the pair of equations below graphically, by substitution and using determinants:

2x – 3y = 8,  

3x + y = 1

Graphically:

x =  1,  y = -2

By substitution:

Make y subject of lower eqn.

y = 1 - 3x

Substitute for y in top eqn.

2x - 3(1 - 3x) = 8

Þ

2x - 3 + 9x  = 8

11x = 11

x = 1

y = 1 - 3x = 1 - 3(1)= 1 - 3 = -2

x = 1, y = -2

By Determinants: For main (D):

a1 =2, b1 = -3, c1 = 8

a2 = 3, b2 = 1, c2 = 1

Þ

Then: D = (2) (1) - (3)(-3) = 2 +9 = 11

Then:

Substituting determinant values: x =

= (8) (1) - (1)(-3) / 11

= 8 - (-3) / 11 =  11/11 = 1

And:

Substituting determinant values:  y =

= (2) (1) - (3)(8) / 11

= (2 - 24)/ 11 =  -22/11 = -2

x = 1, y = -2 

WSJ's Kessler Puzzles Over How U.S. Became Such An Uncivil Nation - Does He REALLY Need An Explanation?

 

Civility died when Trump entered the political landscape

                         Andy Kessler: Clueless as to real source of incivility

America’s descent into chaos and depravity was foretold 10 years ago at a press conference. Donald Trump, who had just been nominated for president at the 2016 Republican National Convention, issued an invitation for one of America’s foes to hack the emails of Hillary Clinton, -  Heather Digby Parton, 'Putin Has Trump Right Where He Wants Him', Permalink

In his recent (Oct. 18) WSJ Op-ed ('Have Some Manners, People', p. A16) columnist Andy Kessler asks:

"Is politeness dead? Being polite has been replaced by the incivility of narcissism. People who worry only about themselves: ME, ME, ME.  

....We're overloaded with hate and vitriol. How does this end? How did we get here?'"

Frankly, it's incomprehensible how Kessler can even ask such questions, given I already described the downfall of American civility some 8 years earlier, e.g.

Brane Space: What Happened To Civility? TRUMP!

So why hasn't our WSJ wordsmith processed that apart from nibbling around the edges, i.e.

"Decorum at the top is sorely lacking. Too many are emulating Donald Trump."

And that's the best you've got? No references to how he said (in an Access Hollywood clip) he could 'grab 'em by the pussy', or his mocking a handicapped NY Times reporter?

 Or how this lump of fecal barbarism has incessantly shouted profanities and lies - as well as invitations to get violent - at countless rallies? Including one in which Trump told a Vegas crowd he'd like to punch a demonstrator in the face, then reminisced about earlier days when they'd be carried out on stretchers. 

Can he even see the headlines outside the reactionary Right media orbit? E.g.

No, I don't suppose he can - or won't. Indeed, from the time this orange fungal turd entered U.S. politics, we've seen nothing but chaos, upheaval, national division, racism and hate. But Kessler spends nine tenths of his column on the incivility of others but devotes barely a sentence to Trump's obvious outrages which ushered in "America's descent into chaos and depravity" as Heather Digby Parton wrote in a recent Permalink post.  Kessler then goes on to society's preference for violence, scribbling:

"Society has changed. Our young generation is filled with Luigi Mangione lovers. Students cheer the murder and rape of Nova Festival concertgoers. Eleven jurors thought a woman who murdered her children should be let off and too many people think this is normal."

Well, uh, yeah, but all that's on Trump and his depravity era too. He basically gave violence his stamp of approval. Too many people across the country thought it was normal when Trump said he could shoot somebody on 5th avenue and people would still vote for him. Or when he invited MAGAs at his sundry rallies to beat up protesters.

Or when he barked last year that 6 Dem officeholders "committed seditious behavior punishable by death."

They included:

• Sen. Elissa Slotkin (D-Mich.) — Former CIA analyst and Iraq War veteran
• Sen. Mark Kelly (D-Ariz.) — Former astronaut and Navy veteran
• Rep. Jason Crow (D-Colo.) — Army veteran
• Rep. Chrissy Houlahan (D-Pa.) — Air Force veteran
• Rep. Maggie Goodlander (D-N.H.)
• Rep. Chris Deluzio (D-Pa.) — Navy veteran

What was their seditious 'crime'? Citing the section in the Military's Uniform Code of Justice regarding the right for service men and women to refuse an unlawful order.

So when Kessler complains about these extreme reactions nd behavior he isn't fully processing how they've been with us since Trump's many violent 2016 rallies. 

Then there was Trump's outburst when he invited Iran to 'take out' LA and San Diego, i.e.

Donald Trump

Trump Thinks It’s Funny to Joke About Iran Bombing US Cities Into Oblivion

Why isn't Kessler paying attention to Trump's chronic actions and words as an agent of chaos and malignancy which amounts to a pestilence and has finally numbed much of the nation. The fact is none of Kessler's examples of malign speech or behavior don't hold a candle to the orange traitor's outrages and ongoing transgressions - especially as a convicted felon.

Does he think it's normal for an American President to warn the nation "will burn" if his party (protecting him from any accountability) doesn't win an election? See:

Trump warns America will burn if Republicans lose this election

Kessler ends by writing:

"I'm not pining for a return to the Victorian era, just some politeness and decency that includes respect and nailing a spike into today's rampant narcissism."

Then, Andy, you should at least be pining for an end to the Trump occupation - preferably removal through the 25th amendment.  Basically, the toxicity eroding the nation's civility won't be gone until Trump is, given he's the source of 99 percent of it.

Alas, Kessler's column isn't the only episode of soft-soaping Trump's evil reign of anti-American influence. We see a similar avoidance of reality in John J. Miller's review of a book by Alan Sepinwall on Rod Serling (WSJ, Oct. 17-18, p. C7).  Near the end of the review Miller writes: 

"Serling wouldn't have approved of President Trump. Mr. Sepinwall even indulges in a reference to Hitler. Perhaps he should review the lesson of 'Four O'Clock, a fourth season episode in which a political fanatic denounces people he considers evil and suffers a comeuppance."

But Miller would actually be better served (and enlightened) by reviewing the lesson of the classic Twilight Zone episode about Adolf Hitler returning entitled "He's Alive" (Season 4, Episode 4), 

The story follows an American neo-Nazi (played by Dennis Hopper) who is mentored by the ghostly apparition of Hitler. The end of the episode noting whenever hate, division and racism reign, Hitler "returns” in that country,  via his despicable ideology.  

As for Miller's citation of the episode 'Four O'Clock' it's a big snoozer about a nut case. As one reviewer described it: 

"There is essentially no story here, just a series of interactions driving home the point that the main character is obsessed with his “mission”. It’s 25 minutes of the viewer going “Yeah, we get it”

Maybe Miller and Kessler need a lesson in selective avoidance of reality. Or, perhaps just getting over the fear of a senile, batshit wannabe tyrant.

See Also:

In Trump’s Diplomacy, Insults Are a One-Way Street. Ukraine Is the Latest Example. - The New York Times

Excerpt:

The latest rift with President Volodymyr Zelensky of Ukraine serves as a reminder that President Trump readily lashes out at other world leaders, only to grow outraged if they fight back. The rules of the road when it comes to President Trump’s relations with foreign leaders are clear: He may trash you. But woe unto you if you whack back.

No one knows that better than President Volodymyr Zelensky of Ukraine, who has been through this particular diplomatic wringer more than once

And:

Brane Space: Trump Regrets Political Violence? Don't Freaking Make Me Laugh!

And:

Donald Trump

Apologies to Caligula: Trump Is Out of His Goddamn Mind

And:

Donald Trump

Trump Reigns as Unapologetic, Dark Lord of Misrule

by Robert Becker | February 16, 2025 - 6:12am | permalink

Early on, DJ Trump perceived his talents more suited to play the truculent Lord of Misrule than plausible presidential prospect. Or competent campaigner. Or rational leader. Or trustworthy man. From Birtherism on, Trump gleefully stalked the opposition as a calculated agent of disorder. That’s why today’s scorched earth, “everything is rigged” demolition derby presents nothing new. De-legitimizing rivals, polls, the media, and his own disloyal party, even trustworthy national vote counting, replicates his opening bad faith gambit. Does he now think more fact-free recklessness, let alone again seeding criminal, post-election violence, attracts undecided voters?

» article continues...

And:

Donald Trump

Could Trump, Who Coveted a Nobel Peace Prize, Become the Worst War Criminal in History?

by Connie Peck | September 25, 2026 - 4:46am | permalink

In a highly aggressive 40-minute speech to the United Nations General Assembly on September 22, President Donald Trump once again issued ominous threats to Iran, saying: “I have a very big decision to make. Will a deal be made with Iran that lets them rebuild and create a far greater country than it ever was before, maybe one of the greatest in the Middle East or even the world? Or do I annihilate the Islamic Republic and do it quickly, never giving them a chance to kill or destroy people or countries again? Do I drive them into hell with no chance of survival and no hope of future greatness or generations?”

He implied that his “very big decision” would come after the midterms, suggesting that Iran is waiting to make a deal until then and, if one isn’t forthcoming, then he would choose annihilation.

» article continues...

And:

Did Trump Just Threaten to Annihilate Iran From the UN Podium?

Swoon In Stocks Highlights Close Linkages To Tech That May Mark End Of Good Times

 

     Alarm as shares of cloud computing Co. dive 50% from earlier this year,

The Business and Finance headlines in The Wall Street Journal were peppered with bad news for investors the past week including:

Small-Cap Stocks Hit Hard Amid Surge In Bond Yields

Scrapped IPO Points To Limits Of AI Frenzy

Markets Hit By Tech Selloff

The top story informed us:

"As bond yields march higher, small companies are taking the brunt of the damage.   The Russell 2000 Index which tracks shares of smaller companies, fell for a fifth straight week, its longest weekly losing streak since May, 2022. The index was down by 0.9% this week and 5,7% during a five week span.  Investors are worried that surging Treasury yields will hit small companies where it hurts."

The news from the middle story wasn't much better:

"The AI boom has sent global stock markets soaring this year, with the S&P 500 and Nasdaq both hitting records earlier this month. However, investor enthusiasm for AI's ability to deliver productivity gains has run up against challenges such as the popularity of cheaper AI systems made in China, safety fears after some agents went rogue and a growing backlash to data center construction across the U.S. and elsewhere."

Then there is the parlous information in the last WSJ piece:

"The swoon in stocks the past week highlighted the close linkages between the economy, markets and a relatively small group of technology highfliers. We know there are about 45 S&P 500 companies deemed as having material exposure to artificial intelligence.  Together, those companies have made up about 39% if the PHLX semiconductor index's 14.1% return through Oct. 11 according to  S&P Dow Jones indices."

These results have Ben Inker, co-head of asset allocation at Boston based money manager GMO, worried. By analogy he points to previous manias which didn't end well such as the railroad boom, and the dot.com boom and bust. According to Inker, AI is no different from those frenzies. In his quote in the same WSJ article:

"If AI spending does not continue to grow, that gets tough for aggregate demand."

Recall, Aggregate demand (AD) is the total spending on final goods and services in an economy at a given overall price level and over a specific time period. It can be simplified to a basic formula:

AD = C + I + G + (X - M)

Where:

• C (Consumption): Spending by households on goods and services (e.g., food, rent, healthcare).
• I (Investment): Business spending on capital goods (e.g., machinery, factories, technology upgrades).
• G (Government Spending): Purchases of goods and services by local, state, or national governments (e.g., roads, schools).
• X - M (Net Exports): Total exports (X) minus total imports (M)

Aggregate demand not continuing to grow then, means decreases in all of the above elements, and likely C and I taking the biggest hits.  The graphs of performance of shares over the last week may be a harbinger of what is headed our way. As the article further notes:

"Warning signs started to flash in the trading of bonds tied to data center companies this week, many of which fell sharply. The losses started with the riskiest and most indebted companies, like Coreweave (a cloud computing company).  But by Friday were showing up in data center bonds with investment-grade ratings.

The fears this week trickled through far reaching corners of the market. Riskier and more heavily indebted companies like the cloud computing upstart Coreweave took the brunt of the blow - shares falling 12% on Friday and 25% for the week.  But the selloff swept through the tech sector, hitting everyone from giant hyperscalers Amazon.com and Alphabet, to high flying chip tocks (like Micron)."

So it appears the writing truly is on the wall for a bubble bursting as Bob Inker has indicated. Certainly, if bond yields continue their upward turn and AI company indebtedness increases more than receipts for products - it will be a highly negative omen.

See also:

Brane Space: A.I.-Bubble Angst Spills Into Bond Market - What It Means

And:

A Brief History Of Bear Markets For (Primarily) Younger Investors

And:

A Sobering Comment From Andrew Ross Sorkin On 60 Minutes: "A Crash Is Coming": What's Behind It?

And:

BIS ('Bank of all Central Banks') Connects Dots To Show Proximity Of Global Financial Crash Linked To AI Boom

An In Depth Treatment Of The Landau Contour - And The Role Of the Inverse Laplace Transform. (Part 1)

In this post we take a deep dive beyond the basic presentation of the Landau contour to examine the role not only of the Laplace transform in its  creation but the inverse Laplace transform as well. To that end, given the increased math detail it may be useful to review residues:

Brane Space: The Residue Theorem and Complex Integrals

And contour integration:

Brane Space: Reviewing Contour Integrals

We saw in previous posts on the Landau contour that an initial value equation is required, based on the Poisson equation:

ik (E₁ ) =  -  4pe ò  d ₃ v f ₁ 

And:

 ¶ _t f ₁ / ¶ t  + ik v x  f ₁ =  (e/ m)  E ₁  ¶ _vx f_o )   

Where  ¶ _t f ₁   denotes a total derivative.  For temporal variation we must do a Laplace transform leading to:

-i w f ₁   -  f ₁   (t = 0) + ik v x  f ₁ =  (e/ m)  E ₁  ¶ _vx f_o 

  The relevant Laplace transform pair being:

A)  E ₁ ( w )  =   ò^¥_o  dt  e ^iwt  E ₁  (t)   And:

B) E ₁ (t)  =  ò_L     dw/2p    e ^-iwt  E ₁  (w)  

Where 'L' for the integral denotes the deformed Laplace contour used in obtaining the inverse Laplace transform.

 The contour L  must pass over all poles of  E ( w ) which from (B) includes all zeros of e (k,w).

Note that E ₁ ( w )  is defined only in the upper half plane of the complex  w- plane.

The preceding pair are based on 2 generic Laplace transforms:

    i) f( w )  =   ò^¥_o  dt  e ^iwt  f(t) 

And:

   ii)  f(t)  =  ò_L     dw/2p    e ^-iwt  f(w)  

In the specific case shown,  E (w ) is defined only for sufficiently large as well as the inverse Laplace transform, defined above by E ₁ (t).  The inverse Laplace transform (B) is accomplished by analytically continuing E (w ) to the entire complex w- plane. By eqn. (A) when one analytically continues E (w ) one must also analytically continue e (k,w). 

e (k,w)  =1 - we ²/ k^{2   ò  du  d} u g(u))/  u   -  w/ k

This  means when w crosses the real w  - axis (from above to below) we cannot allow the pole to cross the integration contour, so we get:

A. Non-deformed contour

If it did cross then the value of  e (k,w)  would jump by -2pi times the residue at u =  w/ k.  Hence the integration contour must be deformed when w < 0    in the complex u-plane.

B. Deformed integration contour

The two sets of contours (A and B) shown in the above graphs are known (in concert)  as the Landau contour. An analogous contour is needed to evaluate the integral :

E (w ) =  4pe/ k² e (k,w)   ò  dv f₁  (k, v, t= 0)/  v x    -  w/ k

which  includes all zeros of  e (k,w). 

With the Laplace contour shown above we have  w _i > 0 everywhere on the contour and hence  wi  > 0 in the evaluation of e (k,w) from the previous equation.

If :

 e (k,w)  = 0  we get a normal mode dispersion relation.

Landau's contribution:

What Landau did was to extend e (k,w) to the entire w- plane. In velocity space the pole can be anywhere in the upper right quadrant (since w _i >  0). To make e (k,w) an entire function one needs to follow an analytic continuation process, i.e. from the upper half plane to the lower half plane.  

For the contour with w _i <  0

Thus we need:

i) I =   ò ^¥_-¥  ^du f_o /  (u   -  w/ k)                              (w >0)

ii) I =   ò ^¥_-¥  ^du f_o /  (u   -  w/ k)  + 2pi (Res)          (w _i > 0)

In part 2 I examine the multiple poles associated with the inverse Laplace transform given by:

E ₁ (t)  =  ò_L     dw/2p    e ^-iwt  E (w)  

And the normal modes associated with them