How to Solve Simultaneous Equations With Squares

By Contributing Writer

Simultaneous equations usually have two or more variables. A solution of such equations is a set of variables that simultaneously satisfies all the expressions. In order to obtain an unambiguous solution, the number of simultaneous equations has to be equal to the number of variables. In the general form, the simultaneous equations that contain variables in squares can be written as: a1 X^2 + b1Y^2 = c1 and a2 X^2 + b2 Y^2 = c2. Abbreviations "a1," "a2," "b1," "b2," "c1" and "c2" are known numeric coefficients in the equations, and "X" and "Y" are variables. As an example, solve the following simultaneous equations: 5 X^2 -- 6 Y^2 = 7 and -4 X^2 + 9 Y^2 = 11.2.

Multiply both sides of the first equation by the coefficient "a2" of the second equation. In our example, 5 x(-4) X^2 -- 6 x (-4)Y^2 = 7 x (-4) or -20 X^2 + 24 Y^2 = - 28.

Multiply both sides of the second equation by the coefficient "a1" of the first equation. In our example, -4 x 5 X^2 + 9 x 5 Y^2 = 11.2 x 5 or -20 X^2 + 45 Y^2 = 56.

Subtract the second transformed equation (Step 2) from the first one (Step 1). Coefficients at the variable "X" are the same after the multiplications in both equations and subtraction will cancel out this term. In this example:

-20 X^2 + 24 Y^2 = - 28 -20 X^2 + 45 Y^2 = 56


(24 -- 45) Y^2 = -28 -56 or -21 Y^2 = -84

Find the solution for the variable "Y." In our example, divide both sides of the expression from Step 3 by "-21" and then take the square root of the quotient. Note that there are two solutions for "Y." (-21/-21) Y^2 = -84/-21 or Y^2 = 4. The solutions are Y = +/-sqrt(4) or Y1 = 2 and Y2 = -2.

Substitute the variable "Y" with its solution from Step 4 in the first equation. Then divide both sides of the equation by the coefficient "a1." In our example, Y^2 is equal to 4. Thus, 5 X^2 -- 6 x 4 = 7 or 5 X^2 = 24 + 7. It can be transformed to 5/5 X^2 = 31/5 or X^2 =6.2.

Take the square root in the expression from Step 5 to find two solutions for the variable "X." In this example, the solutions are X = +/-sqrt(6.2) or X1 = 2.49 and X2 = -2.49. Note that both solutions are rounded to hundredths.

Combine "X" and "Y" values to obtain solutions for the simultaneous equations. In our example, there are four solutions: (2.49, 2), (2.49, -2), (-2.49, 2) and (-2.49, -2).