Lesson OVERVIEW of lessons on solving systems of non-linear equations in two or more unknowns

Source code of 'OVERVIEW of lessons on solving systems of non-linear equations in two or more unknowns'

This Lesson (OVERVIEW of lessons on solving systems of non-linear equations in two or more unknowns) was created by by ikleyn(52781)

: View Source, Show
About ikleyn:

<H2>OVERVIEW of lessons on solving systems of non-linear equations in two or more unknowns</H2>

My lessons on solving systems of non-linear equations in two or more unknowns in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-algebraic-equations-of-high-degree.lesson>Solving algebraic equations of high degree</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-the-system-of-alg-eqns-of-deg2-deg1.lesson>Solving systems of algebraic equations of degree 2 and degree 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-the-system-of-algebraic-equations-of-degree-2.lesson>Solving systems of algebraic equations of degree 2</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Solving-typical-problems-on-systems-of-non-linear-equations-from-the-archive.lesson>Solving typical problems on systems of non-linear equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Some-tricks-to-solve-systems-of-non-linear-alg-eqns.lesson>Some tricks to solve systems of non-linear algebraic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Geometric-word-problems-that-are-solved-using-systems-of-non-linear-equations.lesson>Geometric word problems that are solved using systems of non-linear equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Math-circle-level-problem-on-solving-a-system-of-non-linear-equations.lesson>Math circle level problems on solving systems of non-linear equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Solving-some-special-systems-of-non-linear-algebraic-equations.lesson>Solving some special systems of non-linear algebraic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/The-system-of-nonlinear-alg-equations-with-symm-functions-in-the-left-side.lesson>Solving systems of non-linear algebraic equations with symmetric functions of unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-two-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in two unknowns using the Cramer's rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-three-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in three unknowns using Cramer's rule</A>

<H3>List of lessons on solving systems of non-linear equations in two or more unknowns with short annotations</H3>

<A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-algebraic-equations-of-high-degree.lesson>Solving algebraic equations of high degree</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Examples 1 - 4. &nbsp;Solve the following polynomial equations of high degree

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x^4+5x^3+6x^2=0}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x^3=8}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x^4-13x^2+36=0}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x^6-16x^3+64=0}}}

<A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-the-system-of-alg-eqns-of-deg2-deg1.lesson>Solving systems of algebraic equations of degree 2 and degree 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Examples 1 - 5. &nbsp;Solve the following non-linear systems of two equations in two unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system((x-3)^2+(y-1)^2=4,
-x+2y=1)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system((x-3)^2+(y-1)^2=4,
-x+2y=2sqrt(5)-1)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system((x-3)^2+(y-1)^2=4,
-x+2y=4)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(y=(x-2)^2-2,
-x+2y=-2)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(xy=1,
-x+2y=-2)}}}

<A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-the-system-of-algebraic-equations-of-degree-2.lesson>Solving systems of algebraic equations of degree 2</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Examples 1 - 4. &nbsp;Solve the following non-linear systems of two equations in two unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2+xy+2y^2=74,
2x^2+2xy+y^2=73)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system((x-1)^2+(y-1)^2=4,
(x-2)^2+(y-2)^2=2)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(y=(x-1)^2-2,
y=0.5((2x-3)^2-5))}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2+y^2=4,
xy=1)}}}

<A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Solving-typical-problems-on-systems-of-non-linear-equations-from-the-archive.lesson>Solving typical problems on systems of non-linear equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problems 1 - 5. &nbsp;Solve the following non-linear systems of two equations in two unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2+y^2 = 41,
xy = 20)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2+y^2 = 2,
x-y = 4)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system((y-2)^2 = 9(x+2),
9x^2+4y^2+18x-16y = 0)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(9x^2 + 4y^2 + 18x - 16y = 0, 
(x+1)^2 + 2(y-4)^2 = 12)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2=33y+907, y^2=33x+907)}}}

<A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Some-tricks-to-solve-systems-of-non-linear-alg-eqns.lesson>Some tricks to solve systems of non-linear algebraic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problems 1, 2, 3, 4. &nbsp;Solve the systems of non-linear equations

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2 + y^2 - x - y = 18,
xy + 2x + 2y = 26)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2 + y^2 = 144, xy  = 48)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x + y + sqrt(x+y) = 20, x - y + sqrt(x-y) = 12)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x+y=10,x^3+y^3=300)}}}

<A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Geometric-word-problems-that-are-solved-using-systems-of-non-linear-equations.lesson>Geometric word problems that are solved using systems of non-linear equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the value of &nbsp;"k" &nbsp;for which &nbsp;&nbsp;y = kx-2 &nbsp;&nbsp;is a tangent to the curve &nbsp;&nbsp;y^2 = 10x-x^2.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Find the coefficient &nbsp;"k" &nbsp;such that the line &nbsp;&nbsp;kx + y + 1 = 0

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;is tangent to the circle of the radius &nbsp;{{{sqrt(3)}}} &nbsp;centered at the point &nbsp;(2,1).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;For the circle &nbsp;&nbsp;{{{x^2}}} + {{{y^2}}} = {{{4}}} &nbsp;and the line &nbsp;&nbsp;y = {{{mx}}} + {{{4}}}, &nbsp;determine the exact values of the gradient m so that the line:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) &nbsp;&nbsp;is a tangent to the circle;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;b) &nbsp;&nbsp;intersects the circle in two places;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;c) &nbsp;&nbsp;does not intersect the circle.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Determine the value(s) of &nbsp;k &nbsp;such that the circle &nbsp;&nbsp;x^2+(y-6)^2 = 36 &nbsp;&nbsp;and the parabola &nbsp;&nbsp;x^2 = 4ky &nbsp;&nbsp;will intersect only at the origin.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Find the tangent lines to the parabola &nbsp;&nbsp;x^2 = 6y + 10 &nbsp;&nbsp;passing through the point &nbsp;(7,5), &nbsp;which lies outside the parabola.

<A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Math-circle-level-problem-on-solving-a-system-of-non-linear-equations.lesson>Math circle level problems on solving systems of non-linear equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The sum of the squares of two positive real numbers is &nbsp;218 &nbsp;and their difference

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;multiplied by the smaller number equals &nbsp;42. &nbsp;Find the two numbers.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;The sum of cube roots of two real numbers is &nbsp;128,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;while the sum of the reciprocals of their cube roots is &nbsp;2. &nbsp;Find the numbers.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Solve the system of equations

<A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Solving-some-special-systems-of-non-linear-algebraic-equations.lesson>Solving some special systems of non-linear algebraic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Solve the system of equations

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(1/x + 1/y = -1, x^3 + y^3 = 4
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Solve the system of equations

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2/y + y^2/x = 9,
1/x + 1/y = 3/4)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Solve the system

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x^2 + xy + yz + zx = 30,
y^2 + xy + yz + zx = 15,
z^2 + xy + yz + zx = 18)}}}

<A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/The-system-of-nonlinear-alg-equations-with-symm-functions-in-the-left-side.lesson>Solving systems of non-linear algebraic equations with symmetric functions of unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Solve the system of non-linear equations in three unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x+y+z=12,
xy+yz+zx=44,
x^3+y^3+z^3=288)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Solve the system

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x + 2y  + 4z = 9, 
4yz + 2xz + xy = 13,
      xyz      =  3)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Find an ordered triples (x,y,z) of real numbers satisfying the system of equations

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(sqrt(x)+sqrt(y)+sqrt(z)=10,
xy+yz+zx=38,
sqrt(xyz)=30)}}}

<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-two-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in two unknowns using the Cramer's rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problems 1 - 4. &nbsp;Solve the following systems of non-linear equations in two unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system (2/x + 1/y = 11,
3/x - 5/y = 10)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system (4/(x+2) - 3/(y-1) = 1,
1/(x+2) + 2/(y-1) = 1)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system (1/sqrt(x-3) + 5/sqrt(y+2) = 6,
-3/sqrt(x-3) + 2/sqrt(y+2) = -1)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(2^(2+x) + 3^(3+y) = 275, 
2^(3+x) + 3^(2+y) = 145)}}}

<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-three-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in three unknowns using Cramer's rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problems 1 - 3. &nbsp;Solve the following systems of non-linear equations in three unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(1/x + 2/y - 2/z = -2,
5/x + 9/y - 4/z = -3,
3/x + 4/y - 5/z = -3)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(3/(x+2) + 2/(y-1) - 1/(z-3) = 1,
2/(x+2) + 3/(y-1) - 1/(z-3) = 1,
1/(x+2) - 1/(y-1) + 2/(z-3) = 3)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(3/sqrt(x-1) + 1/sqrt(y-2) - 1/sqrt(z-3) = 2,
3/sqrt(x-1) - 1/sqrt(y-2) + 2/sqrt(z-3) = 7,
-3/sqrt(x-1) + 2/sqrt(y-2) - 1/sqrt(z-3) = -2)}}}

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.