Math 10: week 18 – solving by elimination

  • Write both equations in standard form.

  • Make the coefficients of one variable opposites.

  • Add the equations to eliminate one variable.

  • Solve for the remaining variable.

  • Substitute the solution into one of original equations.

  • Write the solution as an ordered pair.

  • Check that the ordered pair is a solution to both original equations.

 

 

Math 10: week 17 – Solving systems of equations (word problems)

Here are clues to know when a word problem requires you to write a system of linear equations:

(i) There are two different quantities involved: for instance, the number of adults and the number of children, the number of large boxes and the number of small boxes, etc.

(ii) There is a value associated with each quantity: for instance, the price of an adult ticket or a children’s ticket, or the number of items in a large box as opposed to a small box.

-Such problems often require you to write two different linear equations in two variables. Typically, one equation will relate the number of quantities (people or boxes) and the other equation will relate the values (price of tickets or number of items in the boxes).


Here are some steps to follow:

1. Understand the problem.

Understand all the words used in stating the problem.

Understand what you are asked to find.

Familiarize the problem situation.

2. Translate the problem to an equation.

Assign a variable (or variables) to represent the unknown.

Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Use substitution , elimination or graphing method to solve the problem.


Example:

The cost of admission to a popular music concert was $162 for 12 children and 3 adults. The admission was $122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?

1 . Understand the problem:

The admission cost for 12 children and 3 adults was $162.

The admission cost for 8 children and 3 adults was $122.

2. Translate the problem to an equation.

Let x represent the admission cost for each child.

Let y represent the admission cost for each adult.

The admission cost for 12 children plus 3 adults is equal to $162.

That is, 12x+3y=162.

The admission cost for 8 children plus 3 adults is equal to $122.

That is, 8x+3y=122.

33 . Carry out the plan and solve the problem.

Subtract the second equation from the first.

Substitute 10 for x in 8x+3y=122.

Therefore, the cost of admission for each child is $10 and each adult is $14.

 

Math 10: Week 16 – Determining the Number of Solutions of linear equations

By graphing a set of linear equations, you can determine whether the system has no solution, one solution, or an unlimited number of solutions.


A system has one solution when (m1) is not equal to (m2).

Ex.

2x+5=y 

-2x+2=y


A system has no solutions when (m1) is equal to (m2) and (b1) is not equal to (b2)

Ex.

2x+5=y

2x-5=y


A system as and infinite number of solutions when (m1) is equal to (m2) and (b1) is equal to (b2)

ex

2x-3=y

2x-3=y


 

Math 10: Week 14 – Slope equations

There are three main forms of linear equations.

Slope-intercept (y=mx+b), Point-slope (yy1=m(xx1)), and Standard (Ax+By=C)


Slope-Intercept: where m is slope and b is the y-intercept

Point slope: where m is slope and is a point on the line

Standard: where AB, and C are constants


Example

A line passes through the points and Find the equation of the line in all three forms listed above.
Two of the forms require slope, so let’s find that first.
slope=m=Δx/Δy
=−5(−2)/5(−4)
=9/-3
=-3
Now we can plug in m and one of the points, to get point-slope form,
Solving for y, we get slope-intercept form, y:
y – 5 = -3(x+5)
y – 5 = -3x – 15
y = -3x – 10
Adding 3x to both sides, we get standard form, Ax + By = C:
y + 3x = -10

Astronomy Wonder Project – Interplanetary Settlement


Interplanetary Settlement refers to the proposed permanent settlement and exploitation of natural resources of astronomical bodies other than Earth. As such, it represents a type of human presence in space that extends beyond human spaceflight and space outpost operations.


Reasons for Human Space colonisation

Survival of human civilization:

The long-term survival of human civilization and terrestrial life is the primary reason for space colonisation. By establishing alternate sites outside of Earth, the Earth’s population, especially humans, may survive natural or man-made disasters on our own planet. Stephen Hawking, a theoretical physicist and cosmologist, has advocated for space colonisation as a method of preserving mankind on two occasions. Hawking warned in 2001 that the human race will become extinct over the next thousand years unless space colonies can be constructed. In 2010, he declared that mankind had two choices: either occupy space over the next two hundred years, or we will face the long-term prospect of extinction.

Vast resources in space:

There are immense resources in space, both in terms of minerals and energy. According to various estimates, the Solar System alone has enough energy and materials to sustain anywhere from several thousand to over a billion times the current Earth-based human population, largely from the Sun itself. Asteroid mining will also play an important role in space colonisation. Asteroids are rich in water and resources for building structures and shielding. To improve space travel, mining and fuel stations can be created on asteroids rather than on Earth.

Alleviating overpopulation and resource demand:

One reason for space colonisation is to offset the effects of global overpopulation, such as resource depletion. If the resources of space were made available for utilisation and suitable life-supporting habitats were developed, Earth’s boundaries of expansion would no longer be defined. Despite the fact that many of Earth’s resources are non-renewable, off-planet colonies might provide the bulk of the planet’s resource needs. The availability of extraterrestrial resources would reduce demand for earthly resources.

Expansion with fewer negative consequences:

Human expansion and technological advancement have typically led in some type of environmental degradation and the loss of habitats and the animals that inhabit them. In the past, expansion has frequently resulted in the displacement of numerous indigenous peoples, with the resultant treatment of these peoples ranging from encroachment to genocide. Because there is no known life in space, this does not have to be a problem, as some supporters of space settlement have pointed out.


Locations

Location is a frequent source of disagreement among advocates of space colonization. Colonization can occur on a physical body planet, dwarf planet, natural satellite, asteroid, or an orbiting one.

Mercury:

Mercury, once assumed to be a volatile-depleted world like our Moon, is now discovered to be volatile-rich, in fact, richer in volatiles than any other terrestrial body in the inner Solar System. The planet also gets six and a half times the solar flux of the Earth/Moon system, making solar energy a tremendously efficient energy source that could be harvested by orbiting solar arrays and transmitted to the surface or exported to other worlds. A Mercury colony, on the other hand, would need extensive protection from radiation and solar flares.

Venus:

The colonisation of Venus has been the topic of numerous works of science fiction, and it is currently being debated both fictionally and scientifically. However, after the discovery of Venus’s highly hostile surface climate, interest has switched mostly to the colonisation of the Moon and Mars, with suggestions for Venus focusing on dwellings floating in the upper-middle atmosphere and terraforming.

Venus has several parallels to Earth that, if not for the inhospitable environment, may make colonisation easier in many ways compared to other viable places. Venus has been dubbed Earth’s “sister planet” due to its similarities and closeness. Venus also poses numerous major hurdles to human settlement. The weather conditions on Venus are daunting: the temperature at the equator is roughly 450 °C, which is greater than the melting point of lead. The surface atmospheric pressure is also at least ninety times stronger than on Earth, which is similar to the pressure felt beneath a kilometre of water. Additionally, water in any form is virtually nonexistent on Venus. The atmosphere is mostly composed of carbon dioxide and lacks molecular oxygen. Furthermore, the visible clouds are made up of caustic sulfuric acid and sulphur dioxide vapour.

The Moon:

Because of its closeness to Earth and lower escape velocity, the Moon has been proposed as a potential settlement location. Abundant ice in some locations may supply a lunar colony’s water demands. However, the Moon’s lack of atmosphere gives no protection from space radiation or meteoroids, therefore lunar lava tubes have been considered as a solution. The Moon’s low surface gravity is also a source of worry, since it is uncertain if 1/6g is sufficient to sustain human health for lengthy periods of time. Interest in creating a moon base as a gateway to Mars colonisation has grown in the twenty-first century, with concepts such as the Moon Village for research, mining, and trading facilities with permanent inhabitants.

Mars:

Mars colonisation has piqued the curiosity of both governmental and corporate space organisations, and has received significant theoretical portrayal in science fiction literature, cinema, and artwork. Organizations have suggested proposals for a human expedition to Mars, the first stage in any colonisation endeavour, but no one has stepped foot on the planet, and no return journeys have been completed. Landers and rovers, on the other hand, have successfully investigated the planetary surface and given data regarding ground conditions. Interest, the ability for people to offer more in-depth observational studies than unmanned rovers, commercial interest in its resources, and the prospect that the settlement of other planets might reduce the chance of human extinction are all reasons for populating Mars. Difficulties and risks include radiation exposure during the travel to Mars and on its surface, poisonous soil, reduced gravity, the loneliness that comes with Mars’ distance from Earth, a lack of water, and freezing temperatures.

Asteroid belt:

The asteroid belt has a large amount of material accessible, but it is thinly scattered since it covers such a large area of space. Ceres is the biggest asteroid, with a diameter of around 940 kilometres, large enough to be considered a dwarf planet. Pallas and Vesta, both around 520 kilometres in diameter, are the next two biggest. Unpiloted supply spacecraft should be able to reach 500 million kilometres of space with no technical advancement.

Ceres possesses easily available water, ammonia, and methane, which are critical for Mars and Venus’ survival, fuel, and even terraforming. The colony might be developed either on the surface or beneath. However, even Ceres has a minuscule surface gravity of 0.03g, which is insufficient to offset the deleterious consequences of microgravity while making transit to and from Ceres simpler. As a result, either medical treatments or artificial gravity would be necessary. Furthermore, colonising the main asteroid belt would very certainly necessitate the presence of infrastructure on the Moon and Mars.


What is the ideal place for humans?

Mars appears to be our greatest bet for long-term interplanetary colonisation. It has one-third the gravity of Earth, it takes just five to six months to travel there with present technology, and it has vast amounts of ice that might be transformed into liquid water. Solar panels can be easily placed on the planet’s surface as an energy source, and Elon Musk has a not-so-crazy proposal to create fuel on the planet using atmospheric methane.

Although the red planet is not as awful as other locations, it is still a freezing, barren, lifeless wasteland. Certain creatures will not be harmed by this, but humans will still need to construct pressurised dwellings capable of bringing temperatures to a reasonable range and life support systems capable of providing food, water, and oxygenated air as required. There’s also the issue of the planet’s surface being exposed to high levels of UV light and solar radiation in the absence of a substantial atmosphere or magnetic field. And if a solar storm were to strike the planet, it would destroy virtually all critical electronic equipment on the ground and in space.

 

 

 

 

Works Cited

Building a Marsbase is a Horrible Idea: Let’s do it! Perf. Steve Taylor. Kurzgesagt. 2019. Web. 14 May 2022. <https://www.youtube.com/watch?v=uqKGREZs6-w&list=PL_tf9drfAM_1XOzbklaJMC9Y4KBLwSGXM&index=2>.

Colonization of Mars. 9 May 2022. Web. 14 May 2022. <https://en.wikipedia.org/wiki/Colonization_of_Mars>.

Colonization of the Moon. 8 May 2022. Web. 14 May 2022. <https://en.wikipedia.org/wiki/Colonization_of_the_Moon>.

Colonization of Venus. 28 March 2022. Web. 14 May 2022. <https://en.wikipedia.org/wiki/Colonization_of_Venus>.

How To Terraform Venus (Quickly). Perf. Steve Taylor. Kurzgesagt. 2021. Web. 14 May 2022. <https://www.youtube.com/watch?v=G-WO-z-QuWI&list=PL_tf9drfAM_1XOzbklaJMC9Y4KBLwSGXM&index=6>.

How We Could Build a Moon Base TODAY – Space Colonization 1. Perf. Steve Taylor. Kurzgesagt. 2019. Web. 14 May 2022. <https://www.youtube.com/watch?v=NtQkz0aRDe8&list=PL_tf9drfAM_1XOzbklaJMC9Y4KBLwSGXM&index=1>.

Patel, Neel V. THE BEST EXTRATERRESTRIAL WORLDS TO COLONIZE IN THE SOLAR SYSTEM. 5 September 2017. Web. 14 May 2022. <https://www.inverse.com/article/30832-6-best-extraterrestrial-worlds-colonize-solar-system>.

Space colonization. 27 April 2022. web. 14 May 2022. <https://en.wikipedia.org/wiki/Space_colonization#Vast_resources_in_space>.

Unlimited Resources From Space – Asteroid Mining. Perf. Steve Taylor. Kurzgesagt. 2021. Web. 14 May 2022. <https://www.youtube.com/watch?v=y8XvQNt26KI&list=PL_tf9drfAM_1XOzbklaJMC9Y4KBLwSGXM&index=4>.

What If Humanity Became an Interstellar Species? What If. 2018. Web. 14 May 2022. <https://www.youtube.com/watch?v=8VzSqYooxmw>.

 

Math 10: week 13 – slope

b is the slope of the line. Slope means that a unit change in x, the independent variable will result in a change in y by the amount of b.

slope = change in y/change in x = rise/run

Slope shows both steepness and direction. With positive slope the line moves upward when going from left to right. With negative slope the line moves down when going from left to right.

Calculating the slope of a linear function

Slope measures the rate of change in the dependent variable as the independent variable changes. Mathematicians and economists often use the Greek capital letter D or Δ as the symbol for change. Slope shows the change in y or the change on the vertical axis versus the change in x or the change on the horizontal axis. It can be measured as the ratio of any two values of y versus any two values of x.


Slope types:

Positive Slope

 

Negative Slope

Undefined Slope

Zero Slope


 

math 10: week 12 – Vertical Line Test

The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation.


Vertical Line Test

The graph of the equation y2 = x + 5 is shown below

By the vertical line test, this graph is not the graph of a function, because there are many vertical lines that hit it more than once. Think of the vertical line test this way. The points on the graph of a function f have the form (x, f(x)), so once you know the first coordinate, the second is determined. Therefore, there cannot be two points on the graph of a function with the same first coordinate. All the points on a vertical line have the same first coordinate, so if a vertical line hits a graph twice, then there are two points on the graph with the same first coordinate. If that happens, the graph is not the graph of a function.

 

Math 10: Week 11 – Relations and Functions

In mathematics, a function can be defined as a rule that relates every element in one set, called the domain, to exactly one element in another set, called the range. For example, y = x + 3 and y = x2 – 1 are functions because every x-value produces a different y-value.

A relation is any set of ordered-pair numbers. In other words, we can define a relation as a bunch of ordered pairs.


How to Determine if a Relation is a Function?

  • Examine the x or input values.
  • Examine also the y or output values.
  • If all the input values are different, then the relation becomes a function, and if the values are repeated, the relation is not a function.

Are the following ordered pairs are functions?

  1. W= {(1, 2), (2, 3), (3, 4), (4, 5)}
  2. Y = {(1, 6), (2, 5), (1, 9), (4, 3)}
  1. All the first values in W = {(1, 2), (2, 3), (3, 4), (4, 5)} are not repeated, therefore, this is a function.
  2. Y = {(1, 6), (2, 5), (1, 9), (4, 3)} is not a function because, the first value 1 has been repeated twice.

 

Math 10: Week 10 – domain and range

The domain and range are defined for a relation and they are the sets of all the x-coordinates and all the y-coordinates of ordered pairs respectively. For example, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then:

Domain = the set of all x-coordinates = {1, 2, 3, 4}

Range = the set of all y-coordinates = {2, 3}

A domain of a function refers to “all the values” that go into a function. The domain of a function is the set of all possible inputs for the function.

The range of a function is the set of all its outputs.