How can I use Excel to solve 3 equations in 3 unknowns? I have known data in the form of 3 different equations. I would like to solve for the 3 unknowns in excel that satisfy the 3 equations

Question

I have 3 equations in 3 unknowns. How can I solve for the 3 unknowns that satisfy the 3 equations?

Answer 1

Use Excel’s Solver add-in to treat the three unknowns as variables and the three equations as constraints.

1. Enter the variables and equations
   1. Pick three cells to hold the unknowns, for example B1, B2, B3 (these are the variable cells).
   2. In three other cells (say C1, C2, C3), enter formulas that compute the left-hand side minus the right-hand side of each equation using B1:B3. For each equation, the formula should evaluate to 0 when the equation is satisfied.
      • Example: if the equation is 2x + 3y – z = 5 and x, y, z are in B1, B2, B3, then in C1 enter =2*B1+3*B2-B3-5.
2. Set up Solver
   1. On the Data tab, in the Analysis group, click Solver.
      • If Solver is not visible, load it first as described in Load the Solver add-in in Excel.
   2. In Set Objective, select a cell that measures the total error, for example C4 with formula =C1^2+C2^2+C3^2 (sum of squares of each equation’s residual).
   3. Choose Min so Solver minimizes this total error.
   4. In By Changing Variable Cells, select B1:B3 (the three unknowns).
   5. Optionally, add constraints if needed (for example bounds on the variables) using Add in the Subject to the Constraints area.
3. Choose solving method and solve
   1. For linear equations, select Simplex LP (LP Simplex) as the solving method.
   2. For nonlinear equations, select GRG Nonlinear.
   3. Click Solve.
   4. If Solver finds a solution, keep the Solver solution when prompted. The values in B1:B3 are the solution to the three equations.

For a strictly linear system, an alternative is to set up the coefficient matrix and use matrix formulas (e.g., MINVERSE and MMULT), but that is not covered in the provided context.

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References:

• Define and solve a problem by using Solver
• Use Goal Seek to find the result you want by adjusting an input value
• Introduction to What-If Analysis