How to Solve 4 Simultaneous Equations
Simultaneous equations are a set of equations containing multiple variables. A set of equations with n variables will require n linearly independent equations to solve for the variables. Therefore, a set of four linearly independent equations will be required to solve for four variables. This set of equations will consist of four equations of the form aiw + bix + ciy + diz = ei where w, x, y and z are the four variables. The values ai, bi, ci and di are the constant coefficients for the ith equation.
Instructions
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1
Ensure the four equations are linearly independent. This means there is no value f such that aif = aj, bif = bj, cif = cj, dif = dj and eif = ej, where i and j represent different equations. For example, if one of the equations is 2w + 2x + 2y + 2z = 7, and another equation is 4w + 4x + 4y + 4z = 14, we can see that the two equations are equivalent by multiplying the first equation through by 2. In this case, the set of four equations could not be solved since they are not linearly independent.
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2
Derive an equivalent equation from the original set of equations that eliminates one of the variables. For two equations, aiw + bix + ciy + diz = ei and ajw + bjx + cjy + djz = ej, there is a value -ai/aj, where aj is not equal to zero. Multiply the equation ajw + bjx + cjy + djz = ej by -ai/aj to get (-ai/aj)ajw + (-ai/aj)bjx + (-ai/aj)cjy + (-ai/aj)djz = (-ai/aj)ej. We can now add the equation aiw + bix + ciy + diz = ei to (-ai/aj)ajw + (-ai/aj)bjx + (-ai/aj)cjy + (-ai/aj)djz = (-ai/aj)ej to get (bi - aibj/aj)x + (ci - aicj/aj)y + (di - aidj/aj)z = (ei - aiej/aj). Note that we have eliminated the w term.
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3
Apply the technique given in Step 2 to derive new equations from the original four equations that eliminate variables. By judicious selection of the equations and the factors to multiply by, we can eventually derive four equations, Aw = E, Bx = F, Cy = G and Dz = H, that contain only one of the variables.
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4
Solve for w, x, y and z. Once we have obtained four equations of the form Aw = E, Bx = F, Cy = G and Dz = H, we can solve for the four variables, as w = E/A, x = B/F, y = G/C and z = H/D.
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