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The Method of Differentials is a method of solving simultaneous equations. It works for certain equations.

Method
Step 1. Write down the equations that are present.

Step 2. Take partial derivatives of one of the equation until one of the terms only has order 1. You now have your first variable.

Step 3. Substitute this into another equation in the system.

Step 4. Take partial derivatives of this equation from step 3 if there are other unknowns. If there is only one unknown, stop here and solve for that other unknown.

Step 5. Repeat steps 2 to 5 for the other equations as necessary until you have solved the system.

Worked example
Consider the system of equations 2x+y=2 and 8x-y^2=8.

Differentiate equation 1 w.r.t. x, such that we have y=0.

Substitute this into the second equation. Now 8x = 8. Therefore x = 1.

Note that 8(1)-0 = 8 and also 2(1) + 0 = 2, this solves the equation.

Doing this the other way round:

8x-y^2 = 8 ==> -2y=0 ==> y =0. Now 2x + 0= 2 ==> x = 1.

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