Math 300
Notes for Section 1.1
1. (Linear Equations). A linear equation in nvari ables x1,x
2,...,x
nis an equation that
can be written in the (standard) form
a1x1+a2x2+···+anxn=b,
where the coefficients a1,a
2,...,a
nand the constant bare real numbers.
2. (Solutions). A solution to a linear equation in nvariablesisasequenceofnreal num-
bers s1,s
2,...,s
nsuch that the equation is satisfied when we substitute x1=s1,x
2=
s2,...,x
n=sn.
3. (Systems of linear equations). A system of mlinear equations in nvariables is a set
of mlinear equations in nvariabl es :
a11x1+a12x2+... +a1nxn=b1
a21x1+a22x2+... +a2nxn=b2
.
.
..
.
..
.
..
.
.
am1x1+am2x2+··· +amnxn=bm.
A solution to such a system is a sequence of nreal numbers s1,s
2,...,s
nthat is simulta-
neously a solution for each equation in the system. To solve a linear system is to find the
set of all solutions to the system.
4. (Method of Elimination). To solve a system of linear equations, we use the method of
elimination, which consists of repeatedly performing the following operations:
(a) Interchange two equations.
(b) Multiply an equation by a nonzero constant.
(c) Add a multiple of one equation to another equation.
5. Theorem (Number of Solutions). A linear system may have no solutions, exactly one
solution, or infinitely many solutions. (If it has no solutions, we say that it is inconsistent .
If it has at least one solution, we say that it is consistent.)
6. (Derive Tip). To plot several planes, say x+y+z=3,x+y−z=1,andx−y+z=2,
first solve each equation for z:z=−x−y+3,z=x+y−1, and z=−x+y+2. Enter
the expression max(−x−y+3,x+y−1,−x+y+ 2) and plot it in 3D-plot window.
