How to Solve Non-Linear Simultaneous Equations

Graphs of the equations used in the example.

Simultaneous equations are two or more equations with multiple variables. A solution of those equations is a set of variables that simultaneously satisfy all equations. The linear equations are generally given as "Y=aX+b," while non-linear equations can be any expressions not described as linear (e.g. "5X^3-7Y^2=21"). "X" and "Y denote equation variables and numbers before variables (e.g. "5" and "-7") are called coefficients.
As an example, we will solve two non-linear simultaneous equations with two variables "X" and "Y". 2X^2+5Y^2=30 and 3X^2-4Y=20.

Things You'll Need

• Calculator

Instructions

1. Identify a variable that is in the same power in the both equation. In our example it would be "X" as it is in the power of "2" in the two equations.

2. Multiply both sides of the first equation by a coefficient from the second equation at the variable identified in Step 1.
   In our example, the coefficient at "X" in the second equation is "3." Thus 3x2X^2+3x5Y^2=3x30 or 6X^2+15Y^2=90.

3. Multiply both sides of the second equation by a coefficient from the first equation at the variable identified in Step 1.
   In our example, the coefficient at "X" in the first equation is "2." It leads to 2x3X^2-2x4Y=2x20 or 6X^2-8Y=40.

4. Subtract the second modified equation (Step 3) from the first modified one (Step 2). Note that coefficients at one variable are the same in the both modified equations and subtraction will cancel out this term.
   In our example it would be
   6X^2+15Y^2=90
   6X^2-8Y=40
   ------------------------
   15Y2^2+8Y=50.
   Finally, add "-50" to the both sides to get it as 15Y2^2+8Y-50=0.

5. Solve the equation with one variable obtained in Step 4. Note that the solution procedure would depend upon a particular equation.
   In our example, we got the quadratic equation "15Y2^2+8Y-50=0" that has two solutions:
   Y1=(-8+sqrt(64-4x15x(50))/15x2=1.57845.
   Y2=(-8-sqrt(64-4x15x(50))/15x2= -2.11178.
   ("Sqrt" is an abbreviation for the square root math operation).

6. Solve one of the initial equations with respect to the variable that is still unknown.
   In our example, such a variable is "X." Add "4Y" to both sides the second equation and then divide by "3."
   X^2=(20+4Y)/3. By taking the square root you would obtain solutions for X
   X=sqrt((20+4Y)/3) and X=-sqrt((20+4Y)/3). Then substitute "Y" with values found in Step 5 to get
   X1=sqrt((20+4x1.57845)/3)=2.9616.
   X2=-sqrt((20+4x1.57845)/3)=-2.9616.
   X3=sqrt((20+4x(-2.11178))/3)= 1.9624.
   X4=-sqrt((20+4x(-2.11178))/3)= -1.9624.

7. Combine variable values derived in Steps 5 and 6 to obtain the solutions of the simultaneous equations.
   Note that in our example, two values of "X" corresponds to each value of "Y." Therefore, these simultaneous equations have four solutions that can be written as "X", "Y" pairs: (2.9616, 1.57845), (-2.9616, 1.57845), (1.9624, 2.11178) and (-1.9624, 2.11178). Graphically it means that the equation plots are crossed in the four points (see figure) having "X" and "Y" coordinates as listed above.

References

• Algebra 1: Practice Workbook With Examples", Houghton Mifflin School, Jun 2001