2022 AMC 10 B
Complete problem set with solutions and individual problem pages
Consider systems of three linear equations with unknowns , , and
where each of the coefficients is either or and the system has a solution other than . For example, one such system is with a nonzero solution of . How many such systems are there?(The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)
- A.
- B.
- C.
- D.
- E.
The sufficient and necessary condition for a homogeneous linear system of equations to have a non-zero solution is that the rank of its coefficient matrix is less than the number of unknowns. In simple terms, it means that among these three equations, at least one must be "redundant" and cannot provide additional constraints to solve the equation. It is an identity that always holds true. Then we can discuss the following cases:
One of the equations has coefficients that are all zeros. For each ordered triple of coefficients in the equations , there are a total of choices. So for this case, we have choices.
None of the equations has coefficients that are all zeros and one of the equations can be obtained by linearly combining the other two equations (without loss of generality, we consider only addition).
Among the three equations, there are two equations that are identical. For this case, we have choices.
In the equation that is a "sum", only two coefficients are 1 and the other one is 0. For example, for the ordered triples of coefficients in the equations we have . For this case, we have choices.
In the equation that is a "sum", all of the three coefficients are 1. For example, for the ordered triples of coefficients in the equations we have . For this case, we have choices.
So in total, we have choices.
