Week 8: Cramer’s Rule

This week’s blog post is on Cramer’s Rule in the lesson Determinants and Cramer’s Rule. This is in the chapter Systems of Equations. 

In this lesson, we reviewed how Determinants worked from last year. Determinants are practically taking two equation like:

x+2y=3

-3x+5y=7

Then they’re put into an equation with Brackets like

_      _

|   1  2  |

| -3  5 |

You then find the determinant by multiplying cross multiply.

then you subtract.

so it would look like

a  b

c  d  

ad-bc

you multiply a times d and then subtract c times b

For Determinents of 3 by 3

It would be

a b c

d e f

g h i

You then draw it again to

a b c a

d e f d

g h i  g

so you multiply just like you did for hte 2 by 2 but by 3. 

To do cramer’s rule you use the determinant to multiply.

You use the originial numbers from the equations but not taking the answers for the equation or the C for the equation. You then find the determinant of that equation. That will be called D

You then replace the first column with the answers in the original equation and then find the determinant. This will be called Dx

You then replace the second column instead with the answers of the original equation. start all over. the answers in the first column would be the original numbers from the original equation. Find the determinant. This will be called Dy.

Do this however many times you need to if there is a third column it will be called Dz.

To Find x: x=Dx/D

To FInd y: y=Dy/D

To FInd z: z=Dz/D

Then the answer will be (x, y, z)

Below is one example of a problem.

That is how you use cramer’s rule.

SEE YOU NEXT WEEK!!!

Week 7: Systems of Equations

We are starting a new chapter, Systems of Equations and Inequalities. The first lesson of the chapter is Systems of Equations. 

This is review from Algebra which is just substitution and elimination. 

There are two tyes of problems in there.

Inconsistent and Consistent.

Incosistent are no solution which are parallel lines or planes

Consistent is the answer which is one solution (independent point) or infinite solutions (dependent line)

To solve Substitution:

1. solve one euqation for one variable

2. substitute into the other equation

3. solve for the variable

4. back substitute to find the second variable

To solve Elimination:

1. Interchange any two equations

2. multiply by a constant number

3. Add one equation to the other to eliminate a variable

It is very simple to solve these types of equations and all you must do is to either substitute or eliminate. 

Here is one problem. 

One problem is when all x, y, and z is equal to 0. That cannot happen. When this happens, you turn z into t and solve for it. 

One example is shown below:

You solve for x with z as t and then so forth.

That’s how you solve systems of equations.

SEE YOU NEXT WEEK!!!

Week 6: Graphs for polar equations

Below are two examples of polar equations created to make an image on a polar graph.

See You Next Post

Week 6: Graphs of Polar Equations

This week we looked at the Graphs of Polar Equations. 

The Polar Equations had many different Graphs

There were circles, lines, spirals, more circles, rose curves and even more.

We more or less looked at the equation for them.

For circles at the origin
r=a where a is the radius

For Lines through the origins
θ=a

For Spirals:
r=aθ

For Circles with the center on the axis
r=asinθ This is on the y-axis
r=acosθ this is on the x-axis a=diameter

Then for Rose curves
r=asinnθ
r=acosnθ
a means the length of the petal
an odd n=n=number of petals
an even n=2n= number of petals
a^2 =how long
n=how many

for a Lemniscate:
r^2 =acos2θ
r^2 =asin2θ
the sin graph is on the x=y axis

For Cardiods
r=a±asinθ
r=a±acosθ

For Limacons:
r=a±bsinθ
r=a±bcosθ
a/b < 1 interior loop
1<a/b<2 dimpled
a/b≥ 2 convex











This is what we did and how you use these polar equations

SEE YOU NEXT WEEK!!!

Week 5: Polar Coordinates

We looked the Polar Coordinate System, which we have never really used before. We have always used the rectangular coordinate system but not the polar.

The Polar Coordinate System uses (r, θ)
r is the direction it goes in a circle and if the θ>0 then it is counterclockwise if it is less then 0 then clockwise

TO convert polar to rectangular
you use x=rcosθ and y=rsinθ

To convert rectangular to polar you use r^2 =x^2 + y^2 and tanθ=y/x

above is an example of a polar equation being put on the graph. There also are four ways of naming in polar coordinates. Above are the four examples.

These equations are also used to solve for the other side in rectangular and in polar.

Here is an example.

It is similar to verifying but not at the same time.

This is the polar coordinate system

SEE YOU NEXT WEEK!!!