Systems of homogeneous linear equations are closely tied to vector spaces. For example, the solutions of a + 3 b + c = 0 4 a + 2 b + 2 c = 0 are given by triples with arbitrary a , b = a / 2 , and c = − − 5 a / 2. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector equation, namely