Solution RREF.M50 Robert Beezer

Let $c,\,t,\,m,\,b$ denote the number of cars, trucks, motorcycles, and bicycles. Then the statements from the problem yield the equations: \begin{align*} c+t+m+b&=66\\ c-4t&=0\\ 4c+4t+2m+2b&=252 \end{align*} We form the augmented matrix for this system and row-reduce \begin{align*} \begin{bmatrix} 1 & 1 & 1 & 1 & 66\\ 1 & -4 & 0 & 0 & 0\\ 4 & 4 & 2 & 2 & 252 \end{bmatrix} &\rref \begin{bmatrix} \leading{1} & 0 & 0 & 0 & 48\\ 0 & \leading{1} & 0 & 0 & 12\\ 0 & 0 & \leading{1} & 1 & 6 \end{bmatrix} \end{align*} The first row of the matrix represents the equation $c=48$, so there are 48 cars. The second row of the matrix represents the equation $t=12$, so there are 12 trucks. The third row of the matrix represents the equation $m+b=6$ so there are anywhere from 0 to 6 bicycles. We can also say that $b$ is a free variable, but the context of the problem limits it to 7 integer values since you cannot have a negative number of motorcycles.