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

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

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

Solving algebraic equations of high degree

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

x%5E4%2B5x%5E3%2B6x%5E2=0

Solving systems of algebraic equations of degree 2 and degree 1

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

system%28%28x-3%29%5E2%2B%28y-1%29%5E2=4%2C%0D%0A-x%2B2y=1%29

system%28%28x-3%29%5E2%2B%28y-1%29%5E2=4%2C%0D%0A-x%2B2y=1%29

system%28%28x-3%29%5E2%2B%28y-1%29%5E2=4%2C%0D%0A-x%2B2y=2sqrt%285%29-1%29

system%28%28x-3%29%5E2%2B%28y-1%29%5E2=4%2C%0D%0A-x%2B2y=4%29

system%28y=%28x-2%29%5E2-2%2C%0D%0A-x%2B2y=-2%29

Solving systems of algebraic equations of degree 2

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

system%28x%5E2%2Bxy%2B2y%5E2=74%2C%0D%0A2x%5E2%2B2xy%2By%5E2=73%29

system%28x%5E2%2Bxy%2B2y%5E2=74%2C%0D%0A2x%5E2%2B2xy%2By%5E2=73%29

system%28%28x-1%29%5E2%2B%28y-1%29%5E2=4%2C%0D%0A%28x-2%29%5E2%2B%28y-2%29%5E2=2%29

system%28y=%28x-1%29%5E2-2%2C%0D%0Ay=0.5%28%282x-3%29%5E2-5%29%29

system%28x%5E2%2By%5E2=4%2C%0D%0Axy=1%29

Solving typical problems on systems of non-linear equations

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

system%28x%5E2%2By%5E2+=+41%2C%0D%0Axy+=+20%29

system%28x%5E2%2By%5E2+=+41%2C%0D%0Axy+=+20%29

system%28x%5E2%2By%5E2+=+2%2C%0D%0Ax-y+=+4%29

system%28%28y-2%29%5E2+=+9%28x%2B2%29%2C%0D%0A9x%5E2%2B4y%5E2%2B18x-16y+=+0%29

system%28x%5E2=33y%2B907%2C+y%5E2=33x%2B907%29

Some tricks to solve systems of non-linear algebraic equations

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

system%28x%5E2+%2B+y%5E2+-+x+-+y+=+18%2C%0D%0Axy+%2B+2x+%2B+2y+=+26%29

system%28x%5E2+%2B+y%5E2+-+x+-+y+=+18%2C%0D%0Axy+%2B+2x+%2B+2y+=+26%29

system%28x%5E2+%2B+y%5E2+=+144%2C+xy++=+48%29

system%28x+%2B+y+%2B+sqrt%28x%2By%29+=+20%2C+x+-+y+%2B+sqrt%28x-y%29+=+12%29

system%28x%2By=10%2Cx%5E3%2By%5E3=300%29

Geometric word problems that are solved using systems of non-linear equations

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

      Problem 2. Find the coefficient "k" such that the line   kx + y + 1 = 0
                          is tangent to the circle of the radius sqrt%283%29

centered at the point (2,1).

Problem 3. For the circle x%5E2

and the line y =

, determine the exact values of the gradient m so that the line:
                              a)   is a tangent to the circle;
                              b)   intersects the circle in two places;
                              c)   does not intersect the circle.

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

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

Math circle level problems on solving systems of non-linear equations

      Problem 1. The sum of the squares of two positive real numbers is 218 and their difference
                          multiplied by the smaller number equals 42. Find the two numbers.

      Problem 2. The sum of cube roots of two real numbers is 128,
                          while the sum of the reciprocals of their cube roots is 2. Find the numbers.

      Problem 3. Solve the system of equations

                     x + yz = 2,    (1)
                     y + zx = 2,    (2)
                     z + xy = 2.    (3)

Solving some special systems of non-linear algebraic equations

Problem 1. Solve the system of equations

system%281%2Fx+%2B+1%2Fy+=+-1%2C+x%5E3+%2B+y%5E3+=+4%0D%0A%29

system%281%2Fx+%2B+1%2Fy+=+-1%2C+x%5E3+%2B+y%5E3+=+4%0D%0A%29

Problem 2. Solve the system of equations

system%28x%5E2%2Fy+%2B+y%5E2%2Fx+=+9%2C%0D%0A1%2Fx+%2B+1%2Fy+=+3%2F4%29

Problem 3. Solve the system

Solving systems of non-linear algebraic equations with symmetric functions of unknowns

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

system%28x%2By%2Bz=12%2C%0D%0Axy%2Byz%2Bzx=44%2C%0D%0Ax%5E3%2By%5E3%2Bz%5E3=288%29

system%28x%2By%2Bz=12%2C%0D%0Axy%2Byz%2Bzx=44%2C%0D%0Ax%5E3%2By%5E3%2Bz%5E3=288%29

Problem 2. Solve the system

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

Solving systems of non-linear equations in two unknowns using the Cramer's rule

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

system+%282%2Fx+%2B+1%2Fy+=+11%2C%0D%0A3%2Fx+-+5%2Fy+=+10%29

system+%282%2Fx+%2B+1%2Fy+=+11%2C%0D%0A3%2Fx+-+5%2Fy+=+10%29

Solving systems of non-linear equations in three unknowns using Cramer's rule

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

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