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obtained by substituting specific values for t. For example, yields the solution , ; and yields the solution
,.
If we follow the second approach and assign y the arbitrary value t, we obtain
Although these formulas are different from those obtained above, they yield the same solution set as t varies over all possible real
numbers. For example, the previous formulas gave the solution , when , whereas the formulas immediately above
yield that solution when .
Solution (b)
To find the solution set of (b), we can assign arbitrary values to any two variables and solve for the third variable. In particular, if
we assign arbitrary values s and t to and , respectively, and solve for , we obtain
Linear Systems
A finite set of linear equations in the variables , , …, is called a system of linear equations or a linear system. A sequence
of numbers , , …, is called a solution of the system if , , …, is a solution of every equation in the
system. For example, the system
has the solution , , since these values satisfy both equations. However, , , is not a
solution since these values satisfy only the first equation in the system.
Not all systems of linear equations have solutions. For example, if we multiply the second equation of the system
by , it becomes evident that there are no solutions since the resulting equivalent system
has contradictory equations.
A system of equations that has no solutions is said to be inconsistent; if there is at least one solution of the system, it is called
consistent. To illustrate the possibilities that can occur in solving systems of linear equations, consider a general system of two
linear equations in the unknowns x and y:
The graphs of these equations are lines; call them and . Since a point (x, y) lies on a line if and only if the numbers x and y
satisfy the equation of the line, the solutions of the system of equations correspond to points of intersection of and . There are
three possibilities, illustrated in Figure 1.1.1:
