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Week 12- Solving Systems Algebraically

This week we learned how to solve systems algebraically with more than one variable.

For example: x+6y=4 and 0=x-4y

In these types of systems you used steps.

  1. Isolate a variable
  2. Plug the equation for that variable into the second system
  3. solve for that variable
  4. plug variable into original equation, solve for the missing variable
  5. Check!

This may not make complete sense in steps, so I’ll demonstrate using the first example.

  1. x+6y=4 can be rearranged to make x=4-6y
  2. next you plus your new equation x=4-6y into your second equation: 0=x-4y which turns into 0=(4-6y)-4y. This becomes y=2/5.
  3. Next, we plug our value for y into our original first equation x+6y=4, which becomes x+6(2/5)=4
  4. Simplifying the equation: x=1.6
  5. Check y=2/5 and x=1.6: x+6y=4 -> 1.6+6(2/5)=4 Yes! 0=x-4y -> 0=1.6-4(2/5) Yes! So we know we did it correctly

Week 11 – Graphing Linear Inequalities in Two Variables

This week we learned how to graph inequalities with two variables. These inequalities are essentially forms of the equation y=mx+b, used to graph lines. The same idea can be applied to graphing the inequalities.

However depending on the inequality the line will either be broken or solid.

solid line = \leq or \geq

broken line = > or <

using the same idea as y=mx+b, we can find the slope, and the y-intercept.

“m” represents our slope, if it’s a whole number it can be read as \frac{n}{1} with “n” being your rise(y), and 1 being your run(x).

For instance: If you were to graph y>x+2, your y-intercept would be 2, and your slope would be \frac{1}{1}

It would look like this:

The area shaded, is figured out by testing x and y coordinates in your equation, until the equation is made true through the coordinates chosen.

Week 10 – Finding the roots in a quadratic equation

\sqrt{3x}+7=x+1

 

When faced with a problem where you have to solve for “x” but aren’t able to get rid of the \sqrt{3x} by squaring both sides right away, you first have to isolate both “x” variables on one side of the equation.

 

 

Now we can get rid of the square root by squaring both sides.

(\sqrt{3x})^2=(x-6)^2

 

3x=x^2-12x+36

On the right side of the equation, x-6 turns into x^2-12x+36 because we have to expand. You aren’t able to just do the math.

Now, since our equation is quadratic we can make our equation equal to zero, and factor it to find our roots.

 

0=x^2-15x+36

 

Lucky for us this equation factors nicely:

$latex 0=(x-12)(x-3)

So, x=12 and/or x=3

 

To check we plug one of our newly found “x’s” into our equation. If one doesn’t work, there’s still a chance that the other one will. ALWAYS check both!

\sqrt{3x}+7=x+1

 

\sqrt{3*3}+7=3+1

 

\sqrt{9}+7=4

 

3+7=4

 

10=4

This one DOES NOT work!!! However, we’ll check x=12 to check if that one does work.

 

\sqrt{3x}+7=x+1

 

\sqrt{3*12}+7=12+1

 

\sqrt{36}+7=13

 

6+7=13

 

13=13

So x=12 is a root, but x=3 is not. Therefore, the equation only has one root.

 

 

 

 

 

 

Week 9-Graphing Quadratic Equations

This week we learned about graphing quadratic equations in factored form. In factored form, we can find the roots, as well as the axis of symmetry. With this information, we can draw a quick graph depicting what our parabola would look like.

ex. Graph the equation y=-2x^2-6x+20

graphing the equation with only the factored form won’t give us a complete parabola, but it will give us enough information to get a rough idea.

y=-2(x^2+3x-10

 

$latex y=-2(x+5)(x-2)

replace the “y” with 0.

0=-2(x+5)(x-2)

rearranging the bracketed “x’s” will give us our roots

 

(x+5) -> x=-5, (x-2) -> x=2

with the x-intercepts, we know that when graphed will be in the format (x,y). But since the x-intercept is where the parabola crosses the x-axis, the y-intercept will equal 0. The x-intercepts will be (-5,0) and (2,0). The axis of symmetry is half way between both roots, so the AOS will be x=-1.5.

Students Lead Huge Rallies For Gun Control Across U.S.

https://www.nytimes.com/2018/03/24/us/politics/students-lead-huge-rallies-for-gun-control-across-the-us.html

This articles acknowledges the plea’s and cries of America’s youth on Gun Control. There is no excuse for the murders of thousands of students that have taken place over the years. America’s young people will no longer stand by and watch as nothing is done to insure their safety, and instead take initiative in ensuring their future. This article describes the events that took place during the March For Our Lives rallies that took place on March 24, 2018, in over 390 of the country’s 435 congressional districts. The author used descriptive language to acknowledge the major movements that occurred that day, including the many speakers and performers.

Week 8-Parabolas

This week we learned about parabolas and how they can be translated on a graph.

A translation is an image of the original parabola, but moved either horizontally or vertically from the original parent equation y=x^2

Depending on the equation, you can determine where the parabola has moved on the number line.

If the coefficient of x^2 changes, the parabola will either stretch or compress. A coefficient less than 1 will cause the parabola to compress, while a coefficient more than 1 will cause it to stretch.

If the coefficient x^2 is negative the parabola will open down, while if it’s positive it will open up.

In the case that your equation looks similar to y=(x+3)^2, with your numbers in brackets, then the “c” will effect whether your parabola translates left or right. If the number is positive, it will translate left, and if it’s negative it will translate right. It does not follow the general idea that negatives go left and positives go right.

Week 7-Using the Discriminant

While solving quadratic equations, it’s useful to take a look at the discriminant.

The discriminant is area under the square root in the quadratic formula:

if you have a quadratic equation (equation equal to zero with 3 distinct parts), you can use the quadratic formula to solve. Depending on the answer, we can figure out whether the equation will have 1,2 or 0 solutions.

~

Let’s find the discriminant of the equation: x^2-3x+6=0

In a quadratic equation, the parent would be ax^2+bx+c

following the parent, in our equation, a=1, b=-3, and c=6

using this information we can plug in the numbers to our equation to find the discriminant.

If b^2-4ac is our equation, we just put our numbers we found in the spots of the letters, and simplify.

b^2-4ac

 

-3^2-4(1)(6)

 

9-4(6)

 

9-24

 

-15

 

using this we know that since the discriminant is -15, the original quadratic equation does not have any solutions, and using the quadratic formula would not work.

Week 6 – Solving Quadratic Equations Using Factoring

This week we learned how to solve a quadratic equation using factoring.

In order to solve the equation the zero product law must be used.

a*b=0, so either a=0 or b=0.

both sides if the equals sign must be equal.

for example: 8(x+2)(x-7)=0

x+2=0

so x=-2

also, x-7=0

so x=7

a quadratic equation has two possible answers. If you plug either of them into the equation, it should work.

8(x+2)(x-7)=0

x=-2

8(-2+2)(-2-7)=0

8(0)(-9)=0

8(0)=0

0=0

to solve any quadratic equation, the product must be 0. The equation cannot work without it.

Why feminism still matters to young people

Link to Article: http://theconversation.com/why-feminism-still-matters-to-young-people-91299

Feminism is the advocacy of women’s rights on the basis of the equality of sexes. Not that women are superior than men, but that we want to be treated equally in every sense. Some people still find it difficult to grasp the idea of an equal world. I personally am an avid supporter of this and choose to educate myself on how to be the best, and most insightful into this subject as possible. The article centers around how young people, especially young women, are using feminism. The #MeToo campaign is an example of how women are standing up and speaking out against abuse and harassment. The author is very descriptive and fact-oriented in a way that is both clarifying and educational.

Week 5-Factoring Polynomials

When factoring polynomials, we can use a system:

Common?

Difference of squares (2 terms)

Pattern (x^2 x #) (3 terms)

Easy 1x^2

Ugly ax^2

or for an easy jingle, Can Divers Pee Easily Underwater.

using these steps we can factor each type of polynomial to its simplest form.

for example:

3y(y+2) – 9(y+2)

because this polynomial has a common factor, y+2, we can substitute it for an unknown variable such as “a”

3ya – 9a

doesn’t that look better?

now we can use the to find the difference of squares because there’s 2 terms. If there are more than two terms you can skip the and go right to P.

3a is common on both terms so we simplify polynomial to

3a(y-3)

now we substitute our y+2 back for “a”.

3(y+2)(y-3)

and you’re done! this polynomial cannot be simplified farther.

For a polynomial with 3 terms, you will not use the D, but skip right to P depending on whether you have 1x^2 or ax^2 will decide if you use the E or the U.

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