We eliminate one variable using row operations and solve for the other. Say that we wish to solve for
$\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If equation (2) is multiplied by the opposite of the coefficient of
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in equation (1), equation (1) is multiplied by the coefficient of
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in equation (2), and we add the two equations, the variable
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ will be eliminated.
Notice that the denominator for both
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is the determinant of the coefficient matrix.
We can use these formulas to solve for
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ but Cramer’s Rule also introduces new notation:
$\text{\hspace{0.17em}}\text{\hspace{0.17em}}D:$ determinant of the coefficient matrix
${D}_{x}:$ determinant of the numerator in the solution of
$x$
$$x=\frac{{D}_{x}}{D}$$
${D}_{y}:$ determinant of the numerator in the solution of
$\text{\hspace{0.17em}}y$
$y=\frac{{D}_{y}}{D}$
The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a quotient of two determinants.
Cramer’s rule for 2×2 systems
Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.
Consider a system of two linear equations in two variables.
If we are solving for
$\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ the
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ column is replaced with the constant column. If we are solving for
$\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ the
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ column is replaced with the constant column.
Using cramer’s rule to solve a 2 × 2 system
Solve the following
$\text{\hspace{0.17em}}2\text{}\times \text{}2\text{\hspace{0.17em}}$ system using Cramer’s Rule.
Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries
down each of the three diagonals (upper left to lower right), and subtract the products of entries
up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.
From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.
Questions & Answers
The average annual population increase of a pack of wolves is 25.
Period =2π
if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic.
Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation
of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15)
it's standard equation is x^2 + y^2/16 =1
tell my why is it only x^2? why is there no a^2?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations