Tuesday, October 13, 2026

Reviewing Algebra II (Conclusion) Solving Simultaneous Equations

 In Part 5 we conclude this series on Review of Algebra II by examining three methods for solving linear simultaneous equations - which form an important part of  any proper Algebra II course.   Three primary modes of solution include: i) By substitution (often the simplest), ii) graphically and iii) using determinants.  Each of which will be shown in the sample problem.

  The last is especially important given determinants actually have their origin in connection with the problem of solving simultaneous linear equations.

A pair of values, that satisfies each of two equations in two unknowns (say one for x, the other for y) is a simultaneous solution of the two equations.  

Take the following simultaneous equation, with unknowns in x, y:

3x + y  =  12 and x - 3y = 14

To solve by substitution, we take the 1st eqn. and solve for y:

y =   12 - 3x

Then subst. this y into the 2nd equation.

x-  3( 12 - 3x) = 14

And solve for x:

x - 36 + 9x = 14

10x  =  14 + 36 =  50

x = 5

Now subst. x = 5 back into 2nd eqn.

x - 3y = 14

Þ

5  -  3y  = 14  Or:  3y = 5 - 14 = -9

Then:  y =  -9 / 3  

y =  -3

The pair can also be solved graphically yielding:


The graphs are seen to intersect at x = 5 and y = -3.

These simultaneous equations can also be solved using determinants and indeed determinants were developed originally to solve such equations. For a generic pair of equations:  

a1x +  b1y =    c1

a2x +  b2y =   c2

We have the main determinant:

Þ

D = a1 b2 -  a2 b1

However, for the solution x we need the ratio of the determinants, i.e.


Where for the numerator:  D2 = c1 b2 -  c2 b1

For the solution to y we have:


The pair of our original eqns.:  3x + y  =  12 and x - 3y = 14

Yields:  a1 = 3,   b1 = 1  and   c1  = 12

a2 = 1,   b2 = -3  and   c2  = 14

Form main determinant from  a1 , b1, a2, b2 :

Þ

=   (3)  (-3) – (1)(1) = - 9 -1 = -10

Solve for x:

Given top determinant: D2 =  c1 b2 -  c2 b1

= (12) (-3) – (14)(1) =  -36 -14 = -50

Then:  x = -50/-10 = 5 

Solve for y:



Given D2 = a1 c2 -  a2 c1

= (3) (14) – (1) (12) =  42 – 12 = 30

Then: y = D2/ D =30/ -10  = -3 

A shorter (but less instructive, i.e. faithful to the origin) method of finding the x,  y solutions entails simply replacing the respective x and y-columns in the main determinant with the constants (i.e. answers, 12 and 14).

Then for the single x- determinant (Dx), after replacing the x-column with 12 and 14:


Þ   (12)(-3) -  14 (1) = -36 -14 = -50

And for the single y- determinant (Dy) ,  after replacing the y-column with 12 and 14:

Þ   (14)(3) -  12 (1) = 42 -12 =  30 


Then the respective solutions are:



Practice Problems:

1) Solve the system below by substitution:

3x - y = 4

5x + 2y = 3


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

2x - y = 10

5x + 2y = 7


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

4x - y = 5

3x + 2y = 12


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

2x – 3y = 8,  

3x + y = 1



Monday, October 12, 2026

Actual (Not Paper) Patriots Ought To Be All In For Ukraine Bombing Russia's Diesel Plants

 

                     Ukrainian drone ready to blow up diesel distillation plant
               After drone strike on atmospheric-vacuum distillation unit
           'Thank you, Donald, for removing the oil sanctions but you must stop Zelensky bombing us'.

Donald Trump is attempting to provide aid and comfort to a mortal enemy of the United States in defiance of the law as passed by Congress expressing the will of the American people. The Congress damn well better make him stop and do what they have directed him to do by law. If he refuses, they need to remove him from office.- WaPo comment

I don't know what dirt Valdimir Putin has on President Donald J Corruption but it must be horrible!!! Makes me grin ear to ear! 😁 And I like seeing Donald Trump upset while bowing down on his knees to Vladimir Putin! He is the worst President in my lifetime and I am 72 years old! WaPo comment

Leave it to DJT to try and put his “little fingers” on the scale to try and benefit Putin; The Only question is ? What does Putin have over D J T  I Stand with Ukraine - WaPo comment

Ukraine needs to knock out ALL of Russia's oil refining capacity. When the Russian economy goes down, the Russian military will collapse, then Putin will end up in a spider hole like Saddam and the vengeful Russian people will hunt him down. FSB archives will be opened up and Trump's treason will have nowhere to hide.- WaPo comment


Yesterday we learned, e.g.

 How Ukraine uses precision drone strikes to hobble Russian oil refineries - Washington Post

Ukraine has been targeting critical components of Russia’s oil industry that are hard to repair. The strikes have crippled Russia’s diesel production and aroused the ire of the American Traitor leader eager to make nice with Putin to buy his diesel to help stave off Americans hit in their pocketbooks. But making deals with Ukraine’s enemy (as well as Europe’s) is what a traitor does.

This was after the orange traitor stunned the world Friday announcing that the U.S. would immediately lift sanctions on Moscow and start buying Russian diesel. (All this after railing against American "communists".) Trump claimed Russia will immediately supply more than 300,000 tons of diesel to the American and global market, followed by an additional 500,000 tons in November and 1 million tons “immediately thereafter.” And: "Russia will deliver an additional 3 million tons within a short period of time”. 

But most energy experts believe this cynical, self -serving move won't make a dime's worth of difference in the short term - meaning to the end of the year. Apart from which we damned well know this move merely rewards Putin and funds his war. Yet the reactionary media (like the WSJ and FOX) still doesn't believe we have a Russian Asset occupying the Oval Office.

Besides, only a goddamned traitor would spout that Ukraine is responsible for driving up diesel and oil prices when the traitor himself launched the war against Iran that incited Iranians to shut down the Strait of Hormuz. Directly leading to the spike in oil, diesel as well as fertilizer prices not to mention the cost in lives.

Thus,  American allies abroad underscored their support for Zelensky to go ahead and keep bombing the Russkie diesel supplies – and ‘devil take the devils. (The U.S traitor and his bosom buddy Putin). The leaders of the European Union, France, Britain, Poland, Finland and Norway spoke by telephone with Zelensky over the weekend to emphasize their backing for him and for Ukraine in continuing its fight against Russia. Let Trump scream and threaten all he wants.

Polish Prime Minister Donald Tusk wrote in a post on X following a call with Zelensky on Sunday.

“I am absolutely certain that nothing and no one will break Ukraine and its President in their fight against Russian aggression,”

You’ve got it because Zelensky and Ukraine are on the right side of this conflict, Trump and Putin in the wrong. So we need Ukraine to keep on bombing the Russian diesel distillation plants, preferably crippling any U.S. chance of making much use of them. Most of us don’t want momentary and minuscule benefits wrought at the hands of American traitors- presidents or not.

Responding to Trump’s call for his ouster, Zelensky said in an interview with ABC News on Sunday that it was up to the Ukrainian people to choose a leader. But he ought to have added that the U.S. Congress should conduct a lawful ouster of Trump via the 25th amendment – or impeachment for his lawlessness and violations of the Constitution.

Thom Tillis (R, NC), meanwhile, said on the NBC’s  Meet the Press:

“I get it that he may be frustrated, but he needs to understand who the good guys are in this disagreement and who the bad guys are, and it’s not even close.”

Noting “that tens of thousands of people have died over this [Russia’s] illegal invasion.”

Let’s be clear here  that Trump’s waiver of Russian sanctions effectively aligns Washington with Moscow and against the bombing Kyiv has carried out against Russian energy infrastructure. Those targeted strikes have damaged Russia’s energy industry, wiping out one percent of the country’s gross domestic product.

Senate Minority Leader Chuck Schumer correctly accused Trump of “doing a deal with the devil” to “help his flailing political fortunes.”

The European Union’s foreign policy chief, Kaja Kallas, slammed the deal as providing Moscow with “more revenues to wage war.”

Of course, Trump’s attacks on Zelensky were welcomed in Moscow, with Russian special envoy Kirill Dmitriev calling it “iconic and historic.”

Yeah, “iconic and historic” that an American President could surrender to a Russian dictator’s will- but of course that same dictator helped Trump get into office in 2016. As former MI6 spy Christopher Steele wrote in his recent book, Unredacted' :

"It is sometimes easy to forget that Putin helped Trump to become President before, in 2016, as the report by special counsel Robert Mueller made clear."

Adding (p. 125):

"Fusion GPS had unearthed numerous connections between Trump and Russia, including his links with alleged Russian organized crime figures - also that Trump had made many trips to Russia starting in the Soviet era."

 Supporting this were literally troves of substantiated material which the Right typically has ignored in Mueller's (2019) Report:. 

www.justice.gov/storage/report.pdf


 Including the indictments of 12 Russian (GRU) agents.


[Read the indictment here.]

Steele Dossier a fake? Think again! As shown in the comprehensive Dossier - The Moscow Project  it is as real as a heart attack. We know Trump stayed overnight at the Moscow Ritz Carlton in 2013 – where the Kompromat pee tape was made, e.g.

James Carville: There Is A Pee Tape 

Further, three experienced Guardian UK journalists (Luke Harding, Julian Borger, Dan Sabbah) reported:

 “The existence of a leaked Kremlin document describing the meeting of the Russian National Security Council in January, 2016 at which Vladimir Putin ‘authorizes a secret spy operation to support a ‘mentally unstable Donald Trump’ in the 2016 presidential election.”

Thankfully, four Republican senators and two Democratic senators have signed onto the statement released yesterday in protesting this abhorrent deal. They said the administration failed to consult with Congress or submit any explanation to justify its decision as required by statute.

The Republican senators who signed onto the statement included Sens. Susan Collins of Maine, Lisa Murkowski of Alaska, Thom Tillis of North Carolina and John Curtis of Utah. The Democratic senators were Sens. Jeanne Shaheen of New Hampshire and Richard Blumenthal of Connecticut.

“The United States should not provide Putin with a financial lifeline while Russia continues to kill Ukrainian civilians. The Administration must immediately reverse course,”

But Trump's desperation to appease his MAGA base and get them cheaper gas and diesel before Nov. 3 is all-encompassing. His  Russkie handlers are keenly aware of his desperation so they know they have him by the balls. Warning from U.S. Senators be damned, the Dotard handlers know their asset will keep doing their bidding and giving Putin all he wants as a Russian asset. Believe it. And believe we have a president who is basically a Manchurian candidate doing the will of an enemy power - with no chance to stop him unless the Dems get control of congress- House AND Senate. 

Bottom line takeaway: Traitor Trump is willing to sacrifice Ukrainian and European security to grab some momentary headlines, i.e. that diesel prices might come down (despite the fact they won't). While appeasing his Russian master and showing he's not ready to have the pee tape released.  At least not before the FBI 302's (7 pages that should have been in the Epstein files.) showing he had sex with an underage girl, e,g.


See Also:

by Thom Hartmann | October 13, 2026 - 5:04am | permalink

— from The Hartmann Report

Is Putin running America through Trump, is Trump just a “useful idiot” for the Russians, or is he the worst traitor America’s seen since the days of Benedict Arnold?

Yesterday morning, Trump took to his Nazi-infested social media site to announce to the world (and a bit earlier to people who give him $100,000 a month for market-moving information) that Russia and Ukraine had agreed to an “energy ceasefire,” telling the world “Both parties have agreed. Do not break it!”

Ukrainian President Zelenskyy responded that the deal was “news for me,” and the Kremlin spokesman said three days earlier that Russia wasn’t interested in any sort of a limited truce like Trump announced. But Trump made an identical announcement back in September, which was also a lie, and the night of his announcement both countries continued to bomb each other’s refineries and power plants.

» article continues...

And:

After Russian Diesel Deal, Trump Faces New Crisis Over Ukraine - The New York Times

Excerpt:

To the Ukrainians and their European supporters, the attacks on Russia’s fuel refineries are Kyiv’s most effective way to respond to Russia’s bombardment of Ukraine’s energy grid. Ukraine is prepared to stop attacking energy infrastructure, Mr. Zelensky has said, if Russia does the same.

Russia has refused such a deal. Nevertheless, to Mr. Trump, Ukraine’s attacks are an affront to a U.S. administration that has continued to provide intelligence and some military support to Ukraine despite the American president’s harsh words about Mr. Zelensky. Diesel prices in particular have become a sore spot for Mr. Trump, causing farmers’ costs to spike just as harvest season arrives and the midterms are weeks away.

“He better damn well stop,” Mr. Trump told reporters on Saturday, referring to Mr. Zelensky and Ukraine’s attacks on Russian refineries, and ignoring Russia’s strikes on the Ukrainian energy grid. “He wants to make problems for the world. He wants to make problems for our farmers, our ranchers.”

Export statistics indicate that the increase in diesel and energy prices stems largely from Mr. Trump’s war with Iran, though Ukraine’s attacks on Russia have also played a role.

And:

Unbelievable: WSJ Editors Rip Trump For NATO Threats - Yet Still Can't Admit He's A Putin Asset

And:

 'Regime Change' - Perhaps The Best Political Expose in 20 years (A Review)

And:

Russia sending drones to Iran, western intelligence says

Moscow close to completing phased deliveries of lethal weapons, food and medicine to Tehran

And:

by Thom Hartmann | March 12, 2026 - 5:18am | permalink

— from The Hartmann Report

`

Eight of our American service members are dead and over 140 wounded because Iran’s military has suddenly gotten really good at targeting our soldiers, Airmen, and Marines. News reports say they’ve been able to hit us with such precision because Russia is using their extraordinary spy satellite, spy plane, and advanced radar capabilities to help Iran’s military.

The Washington Post, which first reported on this, quoted a Russian military expert as saying that Iran is now “making very precise hits on early-warning radars or over-the-horizon radars,” seeming to validate the concern. The article added:

“Iran possesses only a handful of military-grade satellites, and no satellite constellation of its own, which would make imagery provided by Russia’s much more advanced space capabilities highly valuable — particularly as the Kremlin has honed its own targeting after years of war in Ukraine…”

» article continues...

And:

Trump ratchets up attacks on NATO, says U.S. no longer needs alliance


Excerpt:

"President Donald Trump suggested the U.S. might move away from its commitment to defend NATO allies, criticizing them for not supporting the war in Iran. The United States may stop promising to defend its NATO allies should they come under attack, President Donald Trump said Friday, escalating his verbal barrage against the alliance as his frustration grows that European leaders have not significantly contributed to his war against Iran.

From the outset of the war, European leaders have been leery about the legality and wisdom of attacking the Iranian regime, and many of them continue to say that the only long-term way to contain Tehran’s nuclear ambitions is through diplomacy. European officials also face voters who are furious with Trump for suggesting in January that he would seize Greenland — diminishing their appetite for joining in a war of choice the following month."

And:

by Thom Hartmann | February 3, 2026 - 6:25am | permalink

— from The Hartmann Report

The British newspaper Daily Mail is out with a deeply researched investigative report, the result of a long collaboration between columnists Glen Owen and Dan Hodges, along with Mark Hookham (Assistant Editor Investigations), and Daisy Graham-Brown (Investigative Reporter).

It’s shocking in its detail and its implication that Putin has basically owned Trump for years, even before he ran for president in 2016.

They note of last week’s partial (about 50%) Epstein document release:

“The files include 1,056 documents naming Russian President Vladimir Putin and 9,629 referring to Moscow. Epstein even seems to have secured audiences with Putin after his 2008 conviction for procuring a child for prostitution.”

» article continues...

And:

The ‘Putinization’ of US foreign policy has arrived in Venezuela | Venezuela | The Guardian


Friday, October 9, 2026

Belgian Wins Nobel Physics Prize For Ice Cube Neutrino Observatory Research.

 


Belgian physicist Francis Halzen, has been awarded the 2026 Nobel Prize in Physics “for decisive contributions to the IceCube Neutrino Observatory and the discovery of high-energy neutrinos of astrophysical origin.”  The Royal Swedish Academy of Sciences announced the prestigious award in Stockholm on Tuesday, crediting Halzen with a central role in building the IceCube Neutrino Observatory.

This remarkable achievement recognizes Halzen’s audacious idea to turn an enormous volume of naturally occurring Antarctic ice into a telescope for neutrinos, providing an extraordinary window on the universe and some of its most distant and energetic phenomena.

Halzen, a physicist at the University of Wisconsin-Madison, first proposed what ultimately became the IceCube Neutrino Observatory in 1988. The observatory uses thousands of light sensors buried beneath Antarctic ice to capture ghostlike messengers known as neutrinos. These particles can travel vast distances through space and matter, carrying information from high-energy sources in the sky, including black holes and exploding stars. Halzen has served as IceCube’s principal investigator since 2001, leading the decades-long effort that transformed his ambitious idea into reality.

Together, Halzen’s vision and the discoveries made possible by IceCube opened the world to neutrino astronomy giving scientists a fundamentally new way to observe nature, investigate the universe’s most energetic phenomena, and discover new questions about the cosmos

Neutrinos are tiny elementary particles with no electrical charge, but come off as byproducts of nuclear fusion reactions (see top graphic). They're also referred to as "ghost particles:, since they rarely interact with matter. They can travel across enormous cosmic distances almost completely undisturbed. This allows scientists to investigate some of the universe's most energetic and mysterious events.

However, neutrinos occasionally collide with atoms and produce charged particles that generate faint flashes of light. These signals can be picked up by sensitive instruments embedded deep in the clear Antarctic ice. Halzen first proposed using South Pole ice to detect neutrinos in 1988, realizing that the clear ice beneath the South Pole could act as a  neutrino detector, so on the rare occurrence a neutrino strikes an atomic nucleus, it gives off a flash of light that buried sensors can record.

An upper limit on neutrino mass published last year in the journal Science    Disclosing a value of of mν < 0.45 eV at 90% confidence level, i.e.

Given effectively no mass they can cross between planets, stars and human bodies without being altered, making them rare and hard to catch. There are roughly 100 trillion solar and cosmic neutrinos that pass through our bodies every second.

At grander scales, learning more about neutrinos may help cosmologists fill in their ever hazy picture of the universe, including how galaxies originally clustered together and what influences the expansion of the cosmos since the Big Bang.

Neutrinos occur in abundance across the cosmos, created virtually anytime atomic nuclei fuse or fission. But they carry no electric charge and are notoriously difficult to detect. Neutrinos also come in three types, which physicists describe as flavors.  

Let's review again the basis for these neutrino flavor states. If there are three such states: electron, muon and tau, then there must be three different corresponding neutrino masses which we can call: m1, m2 and m3. Further, the three "flavors" are really different superpositions of the 3 basic neutrino mass states.  Moreover, and to make it more complex, we know that quantum interference between mass states means a neutrino originating in one "flavor" can transmogrify to another over its transit.


Because of the oscillations and quantum interference we need to reckon in a "misalignment" between flavor and the basic neutrino masses. This is done by reference to three independent "mixing angles": Θ 12 , Θ 23  and Θ 13. To a good approximation, oscillation in any one regime is characterized by just one Θ ij and a corresponding mass difference, defined:

 D m ij2 = [m j2 - m i2]

Unlike the latest research published in Science, this earlier edition was confined only to mass differences ( D m) not to neutrino mass itself or even upper limits.

As an example, the probability that a muon neutrino of energy E acquires a different flavor after traversing distance L is:

P = sin2 Θ 23  sin2 (
l23)

where
l23 is the energy -dependent oscillation length, given by:

4ħ E c / (
D m 322)

How well do we know the parameters? Atmospheric neutrino observations yield:

 Θ 23  
» 45 degrees, while D m 322 = 0.0024 eV2.


Meanwhile, solar neutrino data yields 
» 33 degrees for Θ12 and  D m 212 = 0.00008 eV2. (Note: ħ is the Planck constant of action divided by 2 π)  If then:


D m 312  =  [D m 212    +  D m 322 ] = 0.00008 eV2 + 0.0024 eV2

We know,
D m 312  =  0.00248
eV2

Oddly, neutrinos can morph from one flavor to another as they move through space and time, a discovery recognized by the Nobel Prize in Physics in 2015. The underlying mechanism that makes these transformations possible, physicists realized, meant that neutrinos must have some mass. Neutrinos are mind boggingly light, and physicists don’t know why.

However, further intense research at the IceCube Neutrino Observatory may finally provide many more answers - especially for the near absence of mass.

See Also:

A Deep Dive Into Neutrino Detection & How It Also Relates To Matter-Antimatter Asymmetry In The Cosmos

And:

Solar Neutrino Breakthrough: Nuclear Fusion In Sun Now Confirmed With Discovery of Solar Neutrinos

And:

 AIP 2026 Nobel Prize Resources